Nukes, for the first civ to build them. The exact details of how they work vary from game to game, but nuclear weapons are always extremely powerful and destructive.
In I and II: The Great Library, which could be acquired early on and gave you a massive tech advantage. Changes in later games make it less overpowered, but it can still be leveraged for some tricky strategies. Like all World Wonders, only one can be built, even if another civ was just about to finish theirs.
I and II also had the Republic and Democracy governments in combination with the "We Love the X" Day and trade mechanics. While other governments simply gave resource gathering bonuses to cities with enough happy citizens, Republic and Democracy governments got those bonuses by default, and happiness allowed for instantaneous population growth instead. Meanwhile, trade caravans could be built and sent to foreign cities for absolutely ludicrous one-time cash windfalls. The upshot of this was that a civilization with these two governments could effectively pour the vast majority of its taxes into luxuries and watch as all of their cities turned into economic powerhouses over a few turns, continuously pump out trade caravans, and use the absurd amounts of money they would soon have in order to rush build any and all units and improvements necessary. The only downside (a Senate that would veto war declarations and try to force your civilization to make peace during negotiations whenever possible) was negated by the fact that the AI was already suicidally aggressive anyway, and that the player could use spies and their ludicrous piles of money to subvert enemy cities even when not at war. "Power Democracy" takes a bit of fine-tuning to pull off correctly, but it'll allow you to run away with the game if you manage it right.
The original Civilization had the Pyramids wonder. With just Masonry technology needed to build them, they allow you to change your government at whim, to any type of government (even if you haven't yet researched it), without an anarchy period. That basically allows democracy (a game breaker government all in itself) about 3000 BC. And then, if you ever need to kick someone's butt but the Congress overrules it, just start a revolution, declare war, and continue enjoying benefits of democracy the very next turn! The only thing you need to watch out for is to never study Communism, which disables the Pyramids effect, and luckily, it sits pretty high up in the science tree and you don't really need it.
There was supposed to be a major disadvantage to Republic and Democracy: you have to accept any peace offers. But there was a very simple counter: just don't ever meet with representatives from other civilizations. In Civilization II, they tried to eliminate this tactic by allowing a civilization to force a meeting with you any time you move a unit next to one of theirs. Since this ability was exercised most commonly after you take an enemy city, the response to this was to eliminate the defenders from several cities, then take over all of them at once.
The Civilization II Pyramids were also a game breaker as they counted as a free Granary in EVERY city you have from when you build it (right at the start of the game) right up to the end. What's that mean in real terms? Double population growth in all cities and it never expires!
There are also a number of game breaking wonders, such as Leonardo's Workshop, which upgrades all your units to more modern equivilents when the necessary technology is researched.
If another civilization is getting ahead of you technologically, you can take over a city (which allows you to steal a tech), move all of your units out of the city, let them take it back, then retake it before they have time to fortify their troops. Rinse, repeat. Add in some tech-stealing diplomats, and you can eliminate several centuries' worth of technological lead in a few turns.
The game had a bug that allowed each Settler to perform any one Settler action each turn (some of them were intended to take more than a dozen turns). You could also, strangely enough, build railroads in the middle of the ocean.
You can produce unlimited Hides caravans and re-home to a city with high trade. With an advanced civilization, this can result in a new tech each turn, and more money than you'll know what to do with. The game designers apparently tried to prevent this by disallowing re-homing of caravans, but forgot to eliminate it from both menus.
Having both Railroad and Explosives completely changed warfare. If you had enough Engineers, you could build a railroad across any distance of grassland or plains in a single turn, then use that railroad to transport units across the railroad (all in the same turn). Your artillery only has one movement point? That won't stop you from using it to attack a city a dozen squares away. You can also use a chain of Transports to move a unit across any distance of water in a single turn.
The UN was a crucial Wonder. Not because it was particularly useful (you could force other civilizations to make peace with you, but that was rather pointless since they'd then just break the treaty a few turns later), but because if another civilization gets it, having them constantly force you to make peace is a hassle.
Ironically, holding the UN makes it easier for a Democracy to wage war. It allows you to bypass the Senate half the time during negotiations, but this in and of itself isn't why you'd want to own it. Rather, the developers programmed the AI to automatically refuse to talk to you if you tried to contact them immediately after taking a city while owning the Wonder. This was intended to keep the player from forcing peace on an AI civ that had just lost territory, but making the effort yourself denied the enemy the chance to force a meeting on unit contact (see above). In effect, you're using two Obvious Rule Patches to cancel each other out.
If you're going for a conquest victory, the Fundamentalism government type makes taking over the world really easy. Under Fundamentalism you never have any unhappy citizens, and buildings that normally increase happiness instead produce gold (and never require maintenance). You can also produce the Fanatics units, and each city can support up to 8 of them for free. The only downside of Fundamentalism is that scientific research is halved, which isn't a very big deal if you've already researched all the late-game tech or if you're rich enough to just buy technologies from other nations.
Civilization III had the Small Wonder "Wall Street". Its effect? Giving you 5% interest on your treasury per turn. After a few turns you had no money issues for the rest of the game, as long as you kept your treasury above zero. It was later fixed with an Obvious Rule Patch, capping the generated income from the building to 50 gold.
The Persians are considered by many to be the most powerful civilization in the game because they possess the Industrious and Scientific traits, which grants them bonuses to both production and research. Additionally, their unique unit, the Immortals, are the single most powerful ancient age offensive unit (it isn't until knights come along that a unit possesses more offensive power). For perspective, they have 4/2/1 attack/defense/movement, while the middle ages Longbowman has 4/1/1. The sheer power of the Immortals makes it easy for the Persians to conquer other civilizations during the ancient age and even well into the middle ages.
The Ottoman Empire has the same traits as Persia and for its unique unit gets a cavalry replacement with 8 attack strength- the same number Tanks defend at. The tech to unlock them comes in the late Middle Ages, when the strongest non-unique unit defends at strength 4. They're expensive, but you won't need to build anything else until the late Industrial era.
The Statue of Zeus World Wonder gives you a free Ancient Cavalry unit every 5 turns. Said unit has 3 attack (when the best Ancient Age defender is the strength 2 Spearman), defends at 2, has 2 movement points, and best of all an extra hitpoint (in a game where the most a standard unit can have is five). Since retreat odds for mounted units are tied to hitpoint totals, they're notoriously resilient as well, both on the attack and defense.
Armies. Not the standard troops you march around the map, but the special unit you can create with a combat-spawned Great Leader. Load them up with three of the same unit for an (invisible) +1 attack/defense/movement bonus, and the combined health of all three units (four with a certain Small Wonder). They also heal incredibly quickly, even in enemy territory (normally impossible without a late-game Small Wonder) and the AI will never attack them unless it has a much more advanced units (as in, two eras ahead) or literally no other targets on the landmass.
Signing a Right of Passage agreement with an AI, then positioning massive armies right outside the other civilization's cities and declaring war. Cue Curb-Stomp Battle, basically circumventing rules on movement in enemy territory. Granted, this will make the entire world refuse to ever sign Right of Passage agreements with you for the rest of the game. Later games fixed this by teleporting all your military units out of enemy territory the moment you declare war.
Civilization IV still has the Pyramid wonder, with pretty much the same effect as in I - it enables all government civics (Civ IV's equivalent of government types). Combined with stone in one's starting area to halve their cost, this changes them from "hellishly expensive" to "doable", and constructing them gives access to the representation civic, which both allows you to increase the population of your largest cities by around 30-50% for the time period you are in and grants each specialist a research bonus (basically meaning you can treat any specialist like a scientist, and scientists themselves are twice as effective).
Some of the Unique Units in Civilization IV also have a vague Game Breaker-esque status. The Incan Quechua is available from the beginning, cheap and cost-effective against archers, making early rushes for the Incas much easier than any other civilization can pull them off. Also, the Roman Praetorian, which is an early Classical Age unit... with an unit strength more common for the Medieval Era.
In every game the AI occasionally beseeches you for gifts or demands you give them tribute, depending on how powerful you are compared to them, and, often, giving into more powerful nations results in them doing it more and more often and gaining a stranglehold on your resources until you become more powerful than them. If you decline, however, the opposing power will become angered and, should you keep doing it, eventually declare war, which is bad if they are more powerful than you. In IV, you can negotiate tribute, essentially attempting to make a trade out of it or lessen the blow... but the game treats this like a normal trade — that is, the game now treats the negotiations as if you started them in the first place, meaning you can exit with no consequences to relationship, basically destroying the entire tribute system completely thanks to an oversight. Since the AI is pretty conniving and cruel when it comes to lording over any advantage in the first place, this seems relatively fair, all things considered.
Before they were nerfed, China was a monster at warmongering. Their special ability, Art of War, lets them gain Great Generals 50% faster, and before nerf they gave a 20% additional bonus to combat ability on top of a normal Great General's 20%. Combine this with the Chu-Ko-Nu, the Chinese unique crossbowman that dealt more damage per hit and could attack twice. Since you gained Great Generals from combat, the Chu-Ko-Nu's rate of fire fed into the 50% additional Great Generals. And those Great Generals would feed the China war machine further with their massive 40% bonus to combat. Swarms of Chinese Chu Ko Nu could strip even the most massive of cities down to nothing in the blink of an eye, and you'd soon have more Great Generals than you knew what to do with, which would promptly go into feeding you Golden Ages. Combined with an excellent Unique Library and China had almost no weaknesses. Later, the Chukonu were nerfed to deal less damage per hit than a regular crossbowman, and the Great General bonus was reduced to 35% combat bonus rather than 40%.
Civilization V gives us Bushido, the Japanese special ability, which causes units to maintain full defense and attack stats even while wounded. If this wasn't overpowered enough, it's possible to combine this with the Populism social policy, which grants a 25% damage bonus to wounded units. In other words, damaging Japanese units in these conditions actually makes them stronger. However, the effect is less useful in the Gods and Kings expansion thanks to the changes to the combat system, and is now actually considered slightly underpowered.
Great War Bombers can be considered this. If one civ reaches Flight even slightly before other civs do, they'll have a decisive advantage for quite some time. At the time Flight is researched there is nothing on the ground or in the sea that can counter them effectively; it's only once other civs reach Flight that they can build Triplanes to intercept bombing runs from GW Bs, but by that time their ground armies have already been blown to dust by the bombers. It doesn't help that ground and sea units with high effectiveness against GW Bs, such as dedicated anti-aircraft guns and destroyers, come some techs later than Flight in the tech tree. The fact that they can be stacked while no other units can gives them a huge advantage.
In particular, the Americans' B-17s after being upgraded to Stealth Bombers are an unstoppable force.
The "reduction in gold cost of items" effects, all combined, are very strong in the right hands, as each absolute value adds up with the others instead of multiplying with them. 25% for going 3 social policies into the Commerce branch, 15% for building Big Ben, and 33% for units 2 social policies into the Autocracy branch means a whopping 73% discount for units (would be 58% if the effects multiplied). A civ with a lot of gold in their coffers will be able to pump out a new unit in every city every turn for quite some time, and, with enough cash flow, can keep generating units as needed. And those bombers and stealth bombers from the previous point? Since they stack, you can buy as many of those as you can afford at once, and use them all on the very next turn. However, Autocracy does require entering the Industrial era, so it can't be done early into the game.
El Dorado, otherwise known as Skill Dorado, gives 500 gold to the first person who finds it. That's enough to buy an instant settler, which is a massive advantage in the early game. Find it as Spain, you get 1000 gold allowing for two instant settlers.
The downside to Gandhi's special ability could (before a patch nerfed these effects) be entirely negated by a social policy and a wonder. With the right buffs, India could have an incredible populace, while maintaining a happiness so high that they would be in an almost constant Golden Age.
Camel Archers and Keshiks. Camel Archers have a ranged strength of 21, which is more than a Cannon has note The cannon has a huge bonus against cities that the camel doesn't, but it's also unable to defend well against melee attacks, and must be set up before firing which amounts to not being able to move and fire on the same turn, and it can retreat after shooting. With four movement points, this means you can kill your enemies with impunity until your enemy gets something that can actually catch them with enough health left to kill. By the Modern era, though, they're demoted to Goddamn Bats. It's the same deal with Keshiks, only they have less strength, move faster and get half again as much experience as a specialty. The Huns' Horse Archers also operate on a similar principle and are available very early, but they're not quite as powerful for their time.
Special mention goes to the Keshiks and Khan combination for the Mongolia, basically Khan, on top of having the effects of the great general is much faster and can heal adjacent units for 15 hit-points a turn. If you combine it with march where the unit heals every turn regardless of action and the Keshiks boost to Khan generation and fast experience, then Mongolia's army basically consists of insanely powerful and regenerating Keshiks on top of always having a great general boost. No wonder Keshiks are always the number one when it comes to best unique unit vote.
Although a very late game occurrence, as soon as you unlock the Giant Death Robot and have sufficient uranium, opposing armies are essentially completely and totally fucked, especially if you combine them with Stealth Bombers. The combat penalty against cities is all well and good, but even with that, a percentage off of 150 combat strength is all but irrelevant, especially if you happen across a civ that's still playing with swords and musketmen (and there's always one). It's entirely possible to blitzkrieg your way across about 10 cities in a few turns if you're canny about placement.
Austria's special ability in the Gods & Kings expansion becomes this once espionage comes into play. When a rival civilization can use spies to flip one of your allied city-states to its side, having the power to outright annex said city-state, putting it permanently into your camp (unless it gets invaded, of course), becomes very useful. A patch released afterwards nerfed it so that you can't annex city-states unless you've been allied with them for 5 consecutive turns. This makes the ability far less useful in multiplayer, since another player can just spend some money on one of the turns and disrupt the alliance.
The "Faith Healers" belief available for religions grants all units +30 HP per turn they spend next to or in a friendly city. For most of the game, this is merely good. Once aircraft enter the picture, it's extremely powerful; aircraft are infinitely stackable within a city, which means any number of them get the healing benefit simultaneously. Other units have to make their way to the target, and don't get the faith healer's bonus until they make it back home, while aircraft always return back to their city after completing their mission. Combined with the air repair promotion, which lets units heal even when attacking, this makes for an entire air force that recovers nearly half of its maximum health every turn. Since there's no "non-air units" qualifier for Faith Healers, this works in practice the same way it does on paper, and can even make spreading your religion a detriment, since enemies following the same religion still get the bonus.
Siam's replacement for the University, the Wat, gives bonus culture along with the university's very strong boost to science production. Note that it is a replacement for the university. In Brave New World, getting the Jesuit Education reformation belief for your religion allows a city to use their faith to build science buildings instantly ("faith buy"). If Siam could get this belief in their cities, they can faith buy a University and construct a Wat in the same city, leading to a science output that puts even Babylon and Korea, the game's resident Science civs, to shame. Firaxis was quick to fix this after Brave New World came out, but the fix allowed any civto faith-buy a Wat with Jesuit Education (and all except Siam could still build a University on top of it). This problem with the fix was also fixed quickly.
Poland, Babylon, Korea, and Maya gets a lot of flak every time someone used it in multi-player and its not hard to see why when their unique ability are some of the most overpowered in the game:
Poland's unique ability, Solidarity, gives a free social policy every time someone advances to the next era. That essentially means seven free social policies assuming the player starts in the ancient era, that gives Poland insane versatility allowing them to go whatever victory they desire. Coupled with the Ducal stable which on top of the stable which gives bonus production from pastures and for mounted units as well as their winged Hussar, i.e. a souped up lancer, is it any wonder that whenever fans vote for best civilization, Poland always gets voted as number one.
Babylon and Korea have one thing in common that makes them much better than almost all other civilization: Science. While the former gives boost to great scientist generation as well as a free great scientist, the latter not only gives tech boost for each time a science building has been built but also gives more science for each specialist and great person improvement. The reason why they are overpowered is that with science, you can unlock more options you want to do with your economy including more buildings, more advanced unit, and lastly score a quick science victory while the other civilization has just learned the agricultural technology. Oh, and how's this for a "coincidence": they're the only two science-based civs in the vanilla game, and they're both DLC. Subtle.
Last but not least, the Mayans where their unique ability, The Long Count, gives a free great person of each kind every time 394 years have passed when the Mayans managed to unlock theology, the unique building, pyramids is just as bad, in addition to giving more faith than shrine, also gives boost to science. Therefore the Mayans are the ideal civilization to not only found a religion but also has the excuse of going wide.
And now, we have the "Beaker Overflow" exploit. At it's worst, one could compound enough "beakers" (or amount of science currently outputted) to 100k+ by the later parts of the Renaissance era; that's enough to buy just about every techs in the game. Even slightly tamer methods yields thousands of free beakers, a not insignificant amount in the same era.
This could actually backfire to the other end of "broken", too. Accidentally overflowing past about 210k science output (achievable only through this exploit or modding, really) will net you a ridiculously large "beaker" deficit, more than enough to halt your tech progress for the rest of the game.
When it comes to coastal domination victories, no other factions demonstrate it better than England with their brilliant unique ability and possesses some of the best unique units in the game. Their ability is to give naval and embarked units 2 movement as well as an extra spy which is already overpowered but combined with their longbowmen and ship-of-the-line units (The former has 3 range which means it can outrange cities, while the latter is a stronger and cheaper version of the frigate which is already an overpowered unit), you can have an insanely fast and powerful unique units that will guarantee you to become the ruler of the sea.
The simplified mechanics provide a number of opportunities to completely dominate the AI even on the most difficult setting. And the Leonardo's Workshop wonder is so overpowered, since it upgrades all your existing units. Given the right circumstances, you can destroy any AI. First, always produce as many cheap, weak units as possible. As you're doing this, follow the path on your technology tree to discover the internal combustion engine. Time the building of Leonardo's Workshop so that it occurs right after you discover the engine and gain the ability to build a tank. All those cheap warriors you've been building since the beginning? They're now tanks. The game is over in 2 or 3 rounds max.