China has the "Romance of the Three Kingdoms" period that was pretty much this trope. In fact, this trope can be easily summed up in just three battles of this time period: Chibi, where western Shu and eastern Wu alliance to battle against northern Wei; Fan Castle, where Wu betrays Shu with the help of Wei to take control of an incredibly important part of central China; and Yi-Ling, where Shu responds to the betrayal by attacking Wu, losing a huge chunk of troops and their leader, and eventually surrendering to Wei, which, after undergoing a mostly bloodless coup and becoming Jin, quickly sweeps up the rest of the empire.
There were numerous other leaders who clashed with at least one of the kingdoms, including Zhang Jiao, Dong Zhuo, Lu Bu, Yuan Shao, and Meng Huo. It was an...interesting period.
Korea had a similar time period which saw three dynasties battling it out, which were also called The Three Kingdoms.
Large chunks of Chinese history follows this trope when there were multiple rulers claiming imperial title and "mandate of heaven." A particularly prolonged yet "stable" 3-4 way conflict took place between 10th-13th centuries. Most of China, by the end of the 10th century, was unified by the Sung Dynasty. However, northern borderlands were controlled by Khitan Liao Dynasty and the western borderlands were controlled by Tangut Western Hsia Dynasty, while the Manchurian lands were in the hands of as of yet divided Jurchen (Manchu) tribes, all of whom were fighting one another. Sung was able to defeat Liao with the aid of Jurchens, who became a unified empire that called itself the Jin, but, after a Sung betrayal, Jin invaded Sung and captured all of northern China...and continuing the 3-way warfare against both the Sung and the Western Hsia. Then Sung made another alliance with some upstart nomads to the north under some guy calling himself Genghis Khan, who would become a party to this conflict until his grandson eventually conquered all the rival empires in China.
Medieval
The Crusades: Roman Catholics vs. Orthodox Christians vs. Muslims, with the Jews caught in the middle. Oh, and the Mongols show up too.
And members of all three sides taking the opportunity to attack rivals within their own faction — occasionally with the help of members of another faction.
It got more complicated with the Ninth Crusade (1271-1272). Abaqa Khan, the monarch of the Ilkhanate, allied with the Crusaders against the Mamluks. He was a Buddhist. On the other hand, the Northern Crusades (or Baltic Crusades) featured various Catholic factions at war with the last remaining Pagan tribes of Northern Europe. Said Catholic factions rivaled each other and did not hesitate to go into open war. For example the long conflict between Poland and the Teutonic Order. Add a number of Orthodox states, mainly Novgorod, getting involved and you have even more complexity than the Middle Eastern Crusades.
Even more complicated. The Mongols who conquered the Russian lands, the Golden Horde, converted to Islam and went to war against their Ilkhanate cousins in Persia.
Mongols were also involved in the Northern Crusades, but against the Crusaders in this case, as the statelets of northern Russia were also vassals of the Golden Horde (and Alexander Nevsky, the Novgorod prince, was a sworn brother of Sartaq, the Mongol Khan (of the Horde).
The Spanish Reconquest was a constant of Christian and Muslim kingdoms fighting each other. When El Cid Campeador conquered Valencia from the Moors and founded his own state, it turned into a clean three way war, El Cid vs. Castile vs. Almoravid Empire, which eventually faded off after his death.
Early modern
Per general rule, any conflict in Europe during The Renaissance was a melée a trois with the Ottoman Empire as a perennial third side. King Francis I of France shocked Europe by breaking this and allying with them.
Before Hernán Cortés reared his bearded head, Mesoamerica was composed by three civilizations hostile or at least indifferent to each other: the Aztec Empire, who ruled most of the land; the Confederacy of Tlaxcala, who was encircled by Aztec territories but still remained free, and the Purépecha Empire, who clashed against the Aztecs from the other side. Cortés achieved his conquest by allying with Tlaxcala while the Purépecha waited to see what happened, and then they allied with him too after he was the winner.
Francisco Pizarro conquered the Inca by gathering many hostile tribes under the Spanish flag, but after all was said and done, a revolt exploded under one of his allies, Manco Inca, followed by another revolt under Pizarro's own lieutenant, Diego de Almagro. The three sides fought each other for years until the Almagrist revolts were taken down, after which Manco's faction remained the only enemy (he had previously tried to ally with Almagrist survivors, only for them to assassinate Manco in an attempt to be pardoned back into the Spanish Empire) until a new Spanish revolt, this time by Gonzalo Pinzarro and Francisco de Carvajal. After Gonzalo's defeat, the late Manco's faction resisted a few years more before being assimilated.
There's a three-way-war in South East Asia in the early 15th century, taking place after the fall of Malacca (part of Malaysia) in the hands of Portuguese conquistadores, between the newly-formed Johore Sultanate, the Portuegese invaders, and the Acheh Empire, over territorial control for Johore. It was in fact referred to as the "Triangle War of 1511" in Malaysian textbooks.
The arrival of the Spanish to East Asia caused them butt heads with the Portuguese for the right to settle and trade in China and Japan, who had their own interests either, and this still doesn't count the Dutch and several tribes and pirate factions. Even after Spain and Portugal were dynastically unified by Philip II, alliances and incidents happened in many combinations.
The Thirty Years' War started as a fairly simple rebellion of Protestant nobles in Bohemia against the Catholic Emperor. In the end, it was a free for all with Catholic France fighting on the same side as Protestant Sweden (which in turn went on sidequests fighting against Denmark and Poland) against the Emperor. Oh and several German petty states switched sides more than once. To mention nothing of individual soldiers or generals, who rose through the ranks, showed signs of Chronic Backstabbing Disorder or were murdered once they had outlived their usefulness. Whoever won that war (many point to France), the common people who got caught in the crossfire sure as hell lost it.
The English Civil War of the 1640s often had the Royalists, Parliamentarians, and "Club Men". The latter were locals who just thought the other two factions should go and fight over the fate of the country somewhere else.
Even better was the contemporaneous wars in Ireland. It was already complicated — with Catholic Irish (who wanted to be ruled by the Pope), Less Catholic Irish (who wanted the king to treat them better), Nationalist Irish (who wanted independence) and English and Scottish settlers (who trusted each other as little as they trusted the natives). When the civil war broke out in Britain most factions then split along King or Parliament lines as well.
The French Revolution which saw the French against more or less everyone at some point, even the US (undeclared naval war). The best example would be the Jacobin Republic, which fought against the British, Austrians and Prussians while fighting their own civil war against both the moderate Federalists/Girondins in the south and the reactionary Vendeans in the west. The Federalists might have been tolerable to the British, the Vendeans were absolutists like the Austrians, and had they ever met they would have shot each other.
The Haitian Revolution: one group wanted to remain loyal to the French Revolution (mostly free blacks, supported by France), one group wanted independence so they could keep things as they were under the monarchy (mostly whites, supported by Britain), and one group wanted independence for a black nation (mostly slaves, supported by Spain). The free blacks won, but when the French Expeditionary Force arrived and started acting like dicks, the free blacks teamed up with the slaves and established an independent Haiti.
The Jidaigeki period. There was a reason why the era is often referred to as "The Age of the Country at War". At some point, every major power in Japan had a battle with someone else and alliances were very shaky indeed.
Basically every war on the North American continent before the 1840s involved Native American tribes that could lead to... complicated allegiances and events.
Early 20th century to World War II
The Russian Civil War: While it's often remembered as a straight-up Reds vs. Whites, there were actually many other factions, such as the Blacks (anarchists), Greens (local peasant militias, sometimes backed by the Social-Revolutionary party), foreign intervention forces, independence-seeking minorities (most notably the Ukrainian People's Republic and the Democratic Republic of Georgia) and many local warlords who shifted their allegiance as it suited them.
Many smaller countries went into the war nominally allied with one side but with the full intention of seizing land or other perks from the weakened victor, even if that victor was ostensibly an ally.
Several countries backed efforts to establish a southern Slavic state (Yugoslavia) purely because it would weaken the Austro-Hungarian Empire or allow a pretext for war.
The "Young Turks" allied with Germany, bringing the Ottoman Empire with them. They then participated in a covert plan to travel the Middle East, fermenting various nations to call for a 'jihad' against the British Empire. To the Germans, this was a tactical plan to force Britain to either expend troops quelling uprisings or lose India, the crown jewel of its empire. The Turks, however, intended to use the holy warcry to found an all-Turk empire of the Middle East. The plan failed on its very first mission to Afghanistan due to a wide array of tactical and diplomatic blunders, as well as the British keeping hefty bribes flowing to Afghanistan. It didn't help that the German agents decided to brew schnapps and get drunk in a country where alcohol was unheard of. Intoxicated European provocateurs flaunting disregard for local religious laws rarely help under-the-table diplomacy.
World War II was one of these, as two separate wars (Japan vs. China in 1937, Germany vs. Poland in 1939) broke out on opposite sides of the globe and eventually merged into a single massive conflict.
Indeed, the Chinese conflict was a straight example. The Japanese came and took over the major cities in the eastern area. In theory, this caused the Communists and the Kuomintang to work together to fight a guerilla war against the Japanese. In practice, the Japanese tried to keep control over the area, while the rebels tried to gain popular support, with all three sides taking pot shots at each other trying to remove the competition.
Germany and China had an excellent relationship prior to World War II. Indeed, Chiang Kai Shek's best troops were equipped and trained by the Germans and German advisors were still planning their operations when the Second Sino-Japanese War broke out—and would continue to do so until 1938 when Hitler agreed to withdraw them. So, for a year or two, the Germans were indirectly fighting the Japanese in China!
After the Germans pulled out, the Soviet Union stepped in as the major supplier of arms (and "advisors") to the KMT government. When the KMT armies (unofficially) clashed with the Communist Chinese forces from 1939 onward, they may well have been doing so with Soviet arms.
Soviets had a complex role in the Chinese communist movement. Cominten (i.e. Soviet) agents in China distrusted Mao and the local communists (though there were Soviet-style Chinese communists, such as the Twenty-Eight Bolsheviks) and often clashed against them, even while they were supposedly fighting together against both the government and various warlord armies.
Chiang, for his part, was only nominally allied with some of the warlords. Zhang Xueliang was none too pleased that Chiang preferred to focus attention on his fellow Chinese communists rather than the Japanese invaders, and took him prisoner until he agreed, on paper, to play nice.
This was also the case in Yugoslavia during World War II where the Chetniks (nationalist partisans), Tito's communists and the German occupiers and their allies, most notably the Croat Ustashe were also trying to gain or retain control of Yugoslavia. Tito's eventual victory was not wholly due to Soviet intervention, setting up his rivalry with Stalin and the 1948 split with Moscow. There were many alliances forged: Italians weren't really happy with NDHnote Nezavisna Država Hrvatska, or "Independent State of Croatia" (they didn't even allow NDH forces to cross to south part of the state), partisans were fighting against any occupiers for communist Yugoslavia, and Chetniks for royalist Yugoslavia, while Germans simply wanted resources from NDH. So Italians often protected Chetniks and Partisans from NDH, and fought them at other times; Chetniks made alliances with Axis forces against Partisans as well with Partisans against Axis forces; Germans and Ustashi fought Partisans constantly, but sometimes allied with Chetniks against them...
Ukrainian Resistance fought at the same time against Polish Resistance, German Army, and Russian Army.
Ditto in parts of Lithuania that were also claimed by the Poles—e.g. around Vilnius.
The vicious nature of the bureaucratic system in Nazi Germany also often resulted in this, especially in the early days of Nazi rule. There was a time when a shooting war between some combination of the SA, SS and the Wehrmacht could have been a real possibility and was a major motivation for the SS to viciously consolidate power in the late 1930s.
When SS General Sepp Dietrich was told of the failed assassination attempt on Hitler in July 1944, the first thing he wanted to know was whether the attempt was carried out by the SS or by the army, in order to figure out which side he should join. Likewise, in the last days of the Third Reich, Heinrich Himmler supposedly thought that he could make a separate peace with the Western Allies, along with his SS, against the rest of the Nazi regime.
Finland's role in the Second World War could be boiled down to "fought almost everybody". The Finns had just ended the Winter War fifteen months prior to the start of Operation Barbarossa and were convinced to ally with Nazi Germany against the Soviet Union, which in turn led to the Soviet Union's allies, Great Britain, to declare war on Finland. When the war turned against Nazi Germany, Finland sued for a separate peace with the Soviet Union, which in turn required the Finns to fight the Germans as per the armistice agreements. Finland entered a mutual co-operation pact with the Soviet Union (though they only provided a little bit of aerial support for one campaign), but were still technically at war with them and the British. They weren't formally at peace until 1947, when the Paris Peace Treaty was ratified.
Cold War up to present day
In many governments, the power is often divided up into three branches, with the idea being that no one branch can get all of the power because it has to compete with two other branches. Apart from America, many Commonwealth countries like the United Kingdom and New Zealand have separation of power where the government is split into 3 branches (parliament, judiciary, and executive). Many Commonwealth countries countries in general also have several parties jockeying for position. As just one example, Canada has had everything from Melees A Trois to Melees A Cinq. The Liberal, Conservative and New Democratic Parties have been competing for decades, with various other parties (Social Credit, Reform, Bloc Quebecois, etc.) being elected to Parliament at any given time.
In the current Sudanese Civil War between the regular Sudanese Armed Forces and the Rapid Support Forces militias, both sides have been attacked by the al-Hilu faction of SPLM-North rebel group in the Nile and Kordofan theaters.
In early days of First Indochina War, there were not only fights between the French army and Viet Minh but also skirmish between French and Chinese. To make the whole thing even more complicated, the whole thing, early during the First Indochina War, French were helped by Japaneseremnants.
The Yugoslav Wars were a three-way melee between the Serbs (Eastern Orthodox, backed by the Russians, Greeks, and other East Europeans), the Croats (Catholic, backed by Western Europeans) and the Bosnians (mostly Muslim, backed by Arabs and other Muslims). Various alliances were formed and many backs were stabbed in the course of the four-year war.
Soviet aid to the KMT government (against Mao's forces) and the earlier hostile role played by Comintern agents against Mao and his followers in the Chinese communist movement were not forgotten by the PRC leaders after 1949. They were among the contributing factors to the breakdown of the relationship between PRC and USSR in the 1950s.
The bizarre 1967 Opium War, where The Remnant of the Kuomitang that escaped from China and hid in Burma started to traffic opium and heroin with help of the CIA. They were kicked out by a Sino-Burmese force and resettled in Laos. In 1967 they entered in conflict with a local drug warlord where they tried to steal 16 tons of Opium from a drug mule caravan and both factions soon started to fight, soon the CIA-backed Laotian general Ouane Rattikone got involved as he used the royal armed force's resources to bomb and attack both sides. Ouane won the conflict and used the 16 ton captured opium to refine it into heroin and sell to the international market.
After the fall of Samuel Doe's government in the First Liberian Civil War in the early 1990s, the various bitterly competing rebel groups turned their guns against each other. Nearly every major party involved, namely the NPFL rebels and the anti NPFL ULIMO militias, suffered major schisms through leadership disputes and traditional tribal/ethnic rivalries. The next phases of the First Liberian Civil War became a tangled, free for all, warlord playground between the main NPFL rebel group, NPFL dissident factions, the ever feuding factions ULIMO militias, and minor regional militias. Alliances between them formed and broke at dizzying speeds. With his enemies bitterly squabbling with one another, NPFL's Charles Taylor was able to divide and conquer them until he emerged victorious in 1998. His extremely violent, often borderline genocidal, purges at the end of the first civil war drove his enemies together in a second uprising. Learning the mistakes they made, Taylor's opponents were able to fight in a cohesively united front and toppled him in the Second Liberian Civil War in the early 2000s.
The Iraq War was a clash between the US-led coalition, the Shia insurgents, and the Sunni insurgents.
The conflict in Mali has alliances changing at the drop of a hat. Broadly speaking, there are three major factions: the military junta in Bamako who want to rule the country, the secularist Tuareg MNLA, who want an independent secular Tuareg republic in the Azawad, and a collection of allied Islamist groups (most notably Ansar Dine, the flagship Tuareg Islamist movement, Al-Qaeda in the Islamic Maghreb, and an AQIM splinter group called the Movement for Oneness and Jihad in West Africa) who broadly agree on the goal of an Islamic government in the region but differ on technical points and objectives. In early 2012, after the junta took power in Bamako, the MNLA formed an alliance with the Islamists, and the victorious alliance created the Islamic Republic of Azawad in the northeastern half of the country. However, since then the alliance has crumbled, and after the French and ECOWAS began to deploy forces to support the junta and bring down Ansar Dine in January 2013, the MNLA agreed to work with the French to throw the Islamists out.
In the Middle East, there is a cold war currently going on between Saudi Arabia, Iran, and Israel.
The situation in Syria evolved into this by 2013 due to infighting among the rebels: the sides were the Syrian Arab Republic government backed by Iran, the "Free" Syrian Army groups backed by the Gulf states and most of NATO, and the Salafist jihadists aligned with Al Qaeda, backed by no state but receiving lots of funds and recruits from private citizens abroad. While Assad was slowly weakening and the FSA was fraying, and foreign fighters from places like Chechnya, Iraq, and Tunisia became more common, the jihadists gained ground.
In the next few years, it became even more complex. The jihadists fractured into two coalitions, one led by al-Qaeda franchises and the other by the Islamic State of Iraq and the Levant, which also operated in Iraq. Iran sent its army to fight directly alongside the Syrian government against all opposition groups, while Russia did the same with its air force. Meanwhile, a coalition led by the United States consisting of most of NATO dropped the bulk of their support to the FSA (while the Gulf states and Turkey continued it), and launched military operations against ISIL while throwing their weight in with the YPG. The YPG was a Kurdish militia that came to prominence in 2013 when it threw both the jihadists and the Syrian government out of the majority-Kurdish regions in the north, and spent most of its time fighting AQ and ISIL rather than the government. So it's effectively five sides. Assad, Iran, Russia, and Iranian-linked militias (e.g. Lebanese Hezbollah) on one; al-Qaeda and its allies on the other; NATO and the YPG on another and aimed primarily at ISIL; the now decentralized FSA on a fourth, and losing even more ground; and the Islamic State of Iraq and Levant all by itself and quite successful.
By 2018-2019, ISIL had mostly been defeated by a combination of the efforts of all the other factions (but mainly the YPG and NATO), yet the conflict still managed to become more complicated. Turkey began an all-out attack on both ISIL and American-backed Kurdish militias (who have been raising trouble in Turkey and pre-invasion Iraq for ages), and trying to make NATO support it, while still backing FSA groups against the Syrian government, while the other NATO countries had dropped all support for the FSA groups due to their associations with al-Qaeda linked groups. Then Israel stepped up its own airstrike campaign in Syria aimed primarily at Iranian and Hezbollah forces, who NATO, despite being allied with Israel and nominally aligned against Iran, have refused to strike directly in favor of focusing on ISIL. However, things were also simplified a bit more by the Syrian government's defeat of the Southern Front; with their destruction, the last significant relatively moderate rebel groups were gone, leaving the remaining FSA groups pretty firmly in the "jihadist" camp, and cooperating in an Enemy Mine situation with the al-Qaeda linked groups (who they still occasionally fight despite their ideologies being nigh-identical) against the Syrian government and YPG.
Other
The term "battle royal" refers to a fight in which three or more combatants fight until only one is left standing.
The term was first coined by the Romans to describe gladiatorial fights involving three or more combatants who would fight until one remained standing or alive. Such battles were considered brutal even by Roman standards, such that early Christians such as Clement of Rome and Ignatius actively campaigned against it to no success.
The battle royal was revived in the arena of boxing in 19th century America, appearing on the undercard of boxing matches. Before the abolition of slavery, such matches featured five or six black slaves who fought blindfolded and bare-knuckled until either one was left standing or until two fighters remained, at which point the blindfolds were taken off and the fight continued as a normal bout. The practice of the battle royal continued long after the abolition into the early 20th century, with the fighters being allowed to keep their winnings.
Nowadays, battle royals occur in Professional Wrestling (see above), with World Championship Wrestling holding the largest battle royals in the industry in the form of the annual 60-man World War 3 pay-per-view events.
Disney Channel, Nickelodeon, and Cartoon Network have been competing for the attention of a generation of preteen children, with higher-level digital networks following suit.
You may hear it referred to as "war between Muslims and Christians". But in actuality, there were countless factions such as the Phalangists, Palestine Liberation Organization, Amal, Druze, Hizbullah and others. Other countries got dragged into it too. An infamous method of execution that happened during the war was "Identity Killings". Every Lebanese person at that time had an I.D Card which listed their sect or religion. The result was that if you were a civilian trying to get through Beirut in your car, you would come across a checkpoint manned by militants. They would demand to see your I.D, and if you happened to be from a sect that they didn't like, they would pull you out of the car and shoot you in the head. Such behavior by all parties involved characterized the war.
Non-Aligned Movement. Adding to the complication was the defection of China, which became a power in its own right opposed to the Soviet Union.