The Aqueduct in Settlers of Catan: Cities & Knights is a potential game breaker. It allows a player to draw any resource they choose if they receive no production on a turn. If a player only has a few numbers to draw from, the Aqueduct is even better, because it allows the player to take advantage of many custom draws, charging for whatever item they wish to build. Combined with a 2:1 port, a player can gun for commodities too, further increasing their advantage. This is taken Up to Eleven when a specific resource in the game is extremely scarce, because the player with the Aqueduct may be one of the only ones who can get it easily.
The collectible dice game Dragon Dice has had so many game breakers in its past that a group of fans formed a company called SFR, Inc., bought the license to the game, and spent the next eight years revising the rules to impose something resembling balance. The biggest change was the removal of the "Charge" attack option, which offered the chance to inflict twice as many casualties on the defending army while suffering a similar increase in one's own losses... fairly balanced by itself, but easily broken with defensive spells such as "Open Grave", which automatically resurrected all of the affected army's dead.
Vampire: The Eternal Struggle (formerly known as Jyhad), a collectible card game originally created by Wizards of the Coast (based on Vampire: The Masquerade by White Wolf Publishing, who now owns V:tES), was remarkably well-balanced until the Ancient Hearts expansion. Along with several minor balance issues, it introduced the action card "Return to Innocence", which allowed one to essentially destroy an opponent in a single action, at the cost of losing an acting high-level vampire card. In itself, overpowered, but not necessarily game-breaking. However, combined with the "Soul Gem of Etrius" card from the Unlimited release, and certain high-level vampires such as "Etrius" vampire from the preceeding Dark Sovereigns expanion, it became far more troublesome. Decks designed to take advantage of this combination were practically unbeatable until the combination was nerfed by a card text errata issued much later.
Fortunately, the game-breaking nature (and lack of challenge) of this combination resulted in very few people actually using it; and in group play, all other players ganging up on anyone who did. Since V:tES never developed the high-stakes tournament play that other WotC games did (like Magic: the Gathering), as well as being balanced primarily for extended group play, rather than high-speed one-on-one combat, players tended to eschew high-powered combos in favour of more creative story- or theme-oriented decks.
Monocle of Clarity: Ask a player a yes or no question which he must answer truthfully. If the question regards a future event, the player MUST play in accordance to that answer. Sounds harmless, right? Step one: Ask player "When I use my monocle next round, will your answer be yes?" Step two: Tailor your next question according to the previous answer. "Will you let my bleeds go unblocked for the rest of the game?" if he said yes, and "Will you block any of my bleeds for the rest of the game?" if he said no. Your opponent is screwed.
A similar technique will allow you to prevent anyone from ever attacking you, to make yourself the only one allowed to vote on any votes, to prevent anyone from introducing any new cards, and to be promptly shown the door and told to never return to the game again.
Another fine example from Wizards of the Coast: in their Star Wars Miniatures game, a Clone Wars version of Obi-Wan called General Obi-Wan Kenobi, or GOWK, was so insanely hard to damage that it was eventually banned due to dominating the tournament scene. The brokenness was compounded in that GOWK came from a starter set, meaning anyone who wanted one could buy the starter (as opposed to ripping through countless random booster packs).
Duel Masters had Bombazar, Dragon of Destiny, one of the few cards that warped the TCG after its introduction (as the TCG was discontinued two sets after its introduction). It was a Fire/Nature hybrid 7 mana creature with 6000 power; a speed attacker (Hast for Magic players), and a double breaker, and when it was summoned, it destroyed all other 6000 power creatures. It then skipped the opponent's next turn, but at the cost of causing its summoner to lose the duel if they hadn't won by then. As a rule, this was usually enough to finish off the opponent, particularly since Bomabazar itself could already eliminate four shields on its own. While the TCG never had a banlist, they did place Bombazar on a watch list.
Soulswap is another notorious example, given that the OCG banned it and tried to make less powerful versions. It's a three-mana shield trigger that switches a creature from the battle zone with one from the mana zone that has the same cost, and since it can be used on either player, on top of allowing one to recover a card that they'd considered lost to mana, it was incredibly versatile.
Slowking from the Neo Genesis set had a Pokémon Power that allowed its user to flip a coin whenever the opponent played a Trainer card, and if that coin was heads, the Trainer card would return to the user's deck without affecting the game. In the Japanese version of the game, this Power could only be used while Slowking was active. When the card was translated to English, however, it was translated incorrectly. The English version of the card not only allowed its owner to use the Power while Slowking was benched, but the power was cumulative, meaning players could flip a coin for each Slowking they had in play every time their opponent played a Trainer card, and if even one were heads, that card would have no effect. While the Japanese version of the card was barely playable (Slowking was not a good attacker, and was easily KO'ed when active), the English version was exceedingly powerful because a player could place one or more Slowking on the bench, prevent the opponent from playing any Trainer cards, and still play a stronger Pokémon as the active Pokémon, which basically shut down the opponent's deck forever.
Sneasel from the Neo Genesis set was simply banned because it was too powerful offensively; by the second turn of the game, its Beat Up attack allowed the user to flip up to six coins (one for each benched Pokemon and one for Sneasel), netting 20 damage per heads, and + 20 more damage due to the two Darkness energies that were on Sneasel in order to use the attack. Along with broken trainers like Computer Search and Professor Oak, this was easy to accomplish, and from the second turn on it was easy to AVERAGE 80 damage per turn; considering the maximum HP for any Pokemon was 120 at the time (and 120 was a rare trait, the average for a fully evolved Pokemon was about 80-90), it's easy to see why this was banned; you simply didn't win if you didn't play your own Sneasel deck. Sneasel has actually been reprinted in a recent set, but it seems that this is really a matter of the rest of the game catching up to it. It's virtually identical to its old version (resistance is now Psychic -20 instead and it has a Fighting x2 weakness), and still a great card, but at 60 HP it's a bit fragile, and there are a number of not-especially-rare cards in the game with more than 120 HP.
Of interesting note is that, while those two cards were not banned in Japan (Slowking wasn't for reasons already stated), the Base Set trainer card Super Energy Removal was banned due to its brokenness. During the time before the first Modified format when it was finally rotated out in the rest of the world, Super Energy Removal was a staple in essentially every single deck and could easily change the flow of an entire game in an instant.
Lysandre's Trump Card from the Phantom Forces set allows all the player to shuffle all cards in their discard pile into their deck, it is one of the very few cards to be banned as it basically allows the player to quickly draw through their deck without any negative effects and increased the length of battles significantly.
One of the most broken strategies possible is to use two cards in tandem: Darkrai Lv. X and a certain Shiftry. Darkrai had one attack that made the strategy work, which put the defending Pokemon to sleep. However, unlike normal sleep, there were two coins flipped, and if both were tails, the Pokemon fainted (both had to be heads for the Pokemon to wake up.). Shiftry's Poke body made it so that all coins flipped were treated as tails.
Warhammer Daemons of Chaos feature the Great Unclean One. For 600 points, it is possible to field a Lord-level mage who has T6, 10 wounds, 4+ armour, 5+ ward, and regeneration. In larger games, you can field several.
Prior to 8th edition, which was designed to nerf them, daemons were arguably the most broken army ever released in the game's history. EVERYTHING in the army had at least 5+ ward save and caused fear (Lord units causing terror). This basically meant the enemy was, unless it was Tomb Kings or Vampire Counts who all have immune to pyschology, going to spend most of the game running scared. The rare case your enemy doesn't run, he's dead anyways, since the magic the army can throw around will wreck, the worst case being an utterly broken spell that a Lord of Change has access to called Glean Magic that causes you force a target enemy wizard to use ALL their spells on targets you chose, and they can't be dispelled, and the icing on the cake being that Glean Magic was incredibly cheap, especially for what it does. And that's not even getting to how all the units in the army were incredibly strong and the only things that wouldn't being running away due to fear or terror will get torn to pieces in melee. There's a reason the 8th edition nerfed this army and why it was so hated for how overpowered it was.
Slann Mage-Priests can be built to nerf the opposing wizard's effect on the game, some of their magic items make miscasts an accepted fact of life, removing the ability to cast with Irresistable Force, forcing stupidity tests every turn. Saurus Old-Bloods can easily be kitted out to crush spammed infantry thanks to the Carnosaur mount.
Lizardmen in general have been bumped one step closer to gamebreaker thanks to 8th edition's broken magic system making Slann one of the few wizards to bypass all the negatives of the new magic system and the added emphasis on large infantry bricks make Saurus Spear Warrior units into one of the nastiest basic troopers in the game. But with the two together you face waves of tough infantry guarding non-squishy wizards that are blasting all of your army into Hell/pieces/ashes. Then they got nerfed pretty heavily in the 8th edition army book; they're still very scary, but you can no longer buy Slann that get free dice for everything, the miscast-guaranteeing effects are gone, and the resistance to miscasting is reduced to an option allowing you to add or remove one point from the roll. Of course, they still have some of the scariest wizards and basic infantry in the game, but that just makes them "tough", rather than "the annihilating fury of vengeful, presumably dead gods" as they were previously.
Vampire Counts character Mannfred Von Carnstein (Lord edition)can raise ridiculous ammounts of undead if left unchecked. How ridiculous? In a single magic phase he can produce 40 zombies if not dispelled or rolling miscasts.
Whats worse is that those 40 Zombies with the Spammfred build is with roughly average to just below average rolls... According to the 7th edition rules.
Tomb Kings and Vampire Counts in general can be considered overpowered for their ability to continuously summon more and more and more cannon fodder to overwhelm their enemies with.
That was really all Tomb Kings had going for them until the newest armybook.
Dark Elf assassins can cause victims to roll a toughness test on 3D6 or die. Considering that except for Daemons most characters have a Toughness of 4, that means trying to roll below 4 on 3 six-sided dice.
And they are Weapon Skill 9 characters for ninety points each.
Dark Elves also have the War Hydra. 175 points for a S5, T5, W5, 7 ATTACKS. Oh and as a monster it also has D6 S5 auto-hits at the end of every combat. And it has a Breath Weapon that's strength is equal to it's number of remaining wounds (which equals a one use only 2d6 auto-hits, which will usually be S5). And it has 2 beastmasters which have 3 S3 armor piercing attacks. And it has a 4+ armor save and regeneration. A single one of these can wreck entire units. And you can take 2 in a 2K game with enough rare points left over to drop a Bolt Thrower.
Most armies in general have a few gamebreakers at any one time, the exceptions being Ogre Kingdoms or Orcs & Goblins.
Ogre Kingdoms now have the Thundertusk. 250 points for an absolutely huge monster (4 attacks, 6 wounds, S6, T6, riders with 3 S4 attacks each), it has Thunderstomp, which means D6 S6 auto-hits per round of combat. It also has a S3(6) stone thrower AND all enemies within 6 inches of it Always Strike Last. One of these at the center of your line means your charging line means your enemy better have the gods of luck on their side, or their entire line is going to collapse.
The current Lore of Life is disproportionately powerful. Let's put it this way: under the 8th edition magic rules, throwing large handfuls of dice at spells is possible but discouraged, because an irresistible force (or, any two dice coming up 6) automatically produces a wild-magic miscast effect, which is likely to lead to something unpleasant happening to the wizard. "Throne of Vines" allows you to discount this on a 2+, or five times in six. And spells are resolved before you roll on the miscast table, so if you get irresistible force on Throne of Vines...yeah. If you think that's nasty, combine it with "The Dwellers Below" from the same lore, which can cripple entire units by killing large numbers of them with no saves allowed. The main balancing factor? The large handfuls of dice needed to cast it are likely to lead to an irresistible force, the resultant miscast and your mage's head exploding. Unless he's able to ignore miscasts on, for example, a 2+...
The successor game, Warhammer: Age of Sigmar, tries to avoid this by not caring about game balance to begin with, to the point where units don't even have points values, but some of this still crept in as early as the changeover PDF's. High Elf Repeater Bolt Throwers, for example, are ambiguously written so it's not hard to read them as firing 72 shots per turn. Also, with no restriction on the units you can bring at all, fielding 36 Dark Elf War Hydras or an entire wheelbarrowful of Nagashes is mechanically fine; the expectation is that it'll be balanced by you being thrown out of the gaming group, which is kind of an awkward basis.
So far averted with Match Play, the point system of Age of Sigmar, so far its been halted as Games-Workshops most balance system to date.
1000 Blank White Cards is entirely made out of cards created by the players. As such, anyone can make potential game breakers. Finding game-breaking combinations of cards is how the game is usually won. In most games, some cards with ridiculously powerful(or boring) effects are banned:
Cards that allow an instant win.
Conversely, cards that allow an instant loss for someone else. Or everyone else. In which case there really isn't any difference.
Cards that give massive numbers of points are occasionally banned, but it isn't very common. This may manifest as a built-in rule to ban cards of a value greater than 1000 or smaller than -1000.
To be fair, this is more a failure of imagination than anything else. As a wise man once said, "sure, you can make a card that gives you a million points, but how long will it be before someone makes a card that forces you to eat it?
Cards that target a specific, named player. Let's say that there are three players, Person 1, Person 2, and Piggy. Any player could make a card that says "the person to your left gets 350 points" but no card could say "Piggy gets 350 points"
Cards that cause lengthy delays or postpone the game are commonly banned, too.
The Three-Dragon Ante supplemental card game for Dungeons and Dragons 3.5 edition had some pretty nasty cards on its own, meant to suck the stakes and your fellow players dry when it comes to gold coinage. The new Emperor's Gambit edition, meant for D&D 4 can be used in conjunction with the original, essentially meant to suck your opponents dry in just a few rounds. Then again, they can do the same to you, but if you are dealt a few unlucky hands...
Agricola has its share of cards popularly seen as broken. Chief offender is the Taster, which you can use to go first in a round any time you feel like it. In a Worker Placement Game, that's essentially an auto-win for a decent player.
The Wet Nurse lets the person who plays it get extra workers faster and more easily than everyone else. Every time someone plays it, they've virtually won. Even relatively new players playing with experienced ones.
The ridiculous lack of balance in early sets of Legend of the Five Rings is kind of understandable; the development of card games was in its infancy, and designers on the whole didn't really have a handle on the whole "balance" thing. Free "corrupt" gold that accelerated your production to a ludicrous degree if you were lucky enough to get a bunch of it on early turns, an event (Iris Festival) that destroyed every single Shadowlands card in play when Shadowlands was one of the playable factions, a 0-cost action (Breach of Etiquette) that, when played on a Lion clan player on the first turn, resulted in them being totally unable to buy any personality in their deck... these were all merely symptoms of the times. By the time Lotus Edition rolled around, there was no excuse. Lotus Edition was an entire arc of Game-Breaker against Game-Breaker: no-risk duel decks that constantly refilled their hands and collected rewards as they sliced your guys down vs. pirate decks that created ludicrous amounts of gold by "raiding" even though they could play any one of their very powerful personalities as a follower for a paltry 3 gold vs. Khol Wall, which became synonymous with "cheese" vs. Ratling decks with exponentially-growing horde armies vs. ninjas with confusing mechanics that never actually worked right, and effects that killed personalities near-unconditionally were cheap and numerous... every game was a test of "Whose ludicrous tech goes off first?" It reached its nadir with the "Test of Enlightenment" set, where many of the personalities included had abilities that might as well have read "Battle: Kill, like, five guys. Don't even bow to do it." The fact that the next block, Samurai Edition, was going to be much weaker than Lotus was considered a major selling point.
A later game from the same manufacturers, Legend Of The Burning Sands, was completely broken out of the box; this was eventually explained by the fact that it included a bunch of cards copied from Legend of the Five Rings without paying attention to how they worked in the new system. The huge breaker? Blacksmith, a card that gave a +2 strength to any character. In Five Rings, where fights were determined by comparing the total Strength on each side, this was a reasonable advantage; but in Burning Sands, where a unit with higher strength was immune from any unit with lower Strength, it was utterly overwhelming and the game boiled down to rushing out djinni armed with swords as quickly as possible. The game died shortly afterwards.
Warmachine/Hordes is generally considered a fairly balanced game. However, most armies in the Mark II edition had at least one example of a Warcaster or Warlock who handily broke the game.
Cygnar brings us the game's most famous example. Epic Victoria Haley. A caster who offers such ridiculous amounts of board control that it's not uncommon to walk away feeling like your opponent is just playing alone while you watch. She has everything from damaging spells that reduce Spd (which stops charges and slams), she can move enemy models into vulnerable positions, she offers one of the best defensive spells against ranged attacks, and just for kicks she can copy any spells cast in her rather immense control range. Just to top it all off, her Feat makes her dominate the board like nobody else. Every enemy model caught in her control range loses either its movement or action AND she gets to decide the order you activate models. A veteran player will know how make you activate your units in such a perfect way that you will feel lucky to be able to attack anything for your entire turn.
Saeryn of the Legion of Everblight has a feat that keeps every warbeast in her battlegroup from being targeted by melee attacks for one round. Which by itself just sounds incredibly powerful, but it really becomes ridiculous in the details. It effectively means that Warbeasts are also immune to free-strikes which are all melee. The Legions specialization in fast, often flying, warbeasts means that they are completely free to position themselves INSIDE enemy ranks without issue. Which synergizes perfectly with one of her spells which creates a large Ao E around a friendly Warbeast which Auto-hits any enemies caught inside.
Epic Lylyth (Shadow of Everblight) is rather infamous for absolutely crushing new players. In any army that largely ignores stealth, she provides Snipe to everything in her battlegroup, and a free additional shot. Combined with Ravagors which have a powerful, already long-ranged attack which leaves a damaging Ao E and continuous fire effects and Bolt Throwers which push targets back. eLylyth's strategy is to completely shred the opponent from across the board while completely ignoring many forms of defense against ranged combat.
It has been nerfed, but the Pirate Queen Skarre was also well known for a technique called 'Skarre Bombing'. Her ability Sacrificial Strike gives her a 10-inch auto-hit attack based on the Armor of a single friendly model that she sacrifices, with her feat also buffing friendly armor by up to 5 (post-nerf it only accounts for base armor), it was easy to end up with an unmissable pow 21 shot. This is a game where a common mantra for players is that even a luck handcannon shot (pow 12) can handily end a game. A pow 21 would devastate all but the hardiest warcasters, if not leave them in a borderline crippled state and easy to finish off.
Outside the realm of Warcasters and Warlocks we have some units and models that are considered quite gamebreaking. For a long time the innocuous Bile Thralls of the Cryx army were absolutely devastating to any unit heavy armies. On paper they seem balanced enough. Cheap, slow, weak, armed only with a pitiful ranged weapon that they are absolutely abysmal at aiming with, their statline is awful beyond awful. However, they possess a large suicide attack that auto-hits anything caught for fairly hefty damage. Getting an entire unit across the board is almost impossible, but one or two should be easy enough, and that's all you need to completely obliterate lower armor units.
The Light Side "Limited Resources" card. Since there was no limit to the number of cards you could have in your hand, and your life depended on having a good reserve of cards in your stack, a card that allows you to move cards from your hand back to the stack can be crucial to winning.
"The Professor", from Dagobah. A defensive shield card that forced any opponent, if they wanted to deploy a card with ability for free, to instead use X Force (where 'x' is half the card's stated deploy cost, rounded up). It essentially forced one's opponent to burn more and more Force each turn, to unreasonable levels. This was a game-nuker that unfairly punished about half the cards in the entire card game, and quickly proved to be exploitable to the extreme.
"Padme Naberrie" from the Episode I set. An incredibly cheap character, she was a breaker in Skywalker-heavy decks, and could be used to deal direct damage to a player based on how many Skywalkers were on the battlefield at one time.
"The Emperor" from Endor. A single-card, non-weapon, out-of-battle character who can bury opposing characters like no one's business (and at any time, no less), with absurd power levels, extra Force generation, and can be further augmented with "helper" cards. The existence of The Emperor (and its impact on breaking the game) forced Decipher to release Mon Mothma to try and counter it (and that in itself became a broken card).
"Krayt Dragon Howl" from Unlimited screwed with the original design philosophy that the Dark Side was intended to go first in every game. Why? It let the Light Side get Force regeneration before the game even started, and can be defended by an Obi-Wan card that was a tank in combat and could pull Disarmed if the opponent tried to take him down with a weapon. Basically, unless a DS player could kill Obi-Wan on the first turn, the LS had already won the game.
It's hard to set up, even assuming the cards are available in the first place, but in Dominion there's a King's Court - King's Court - Goons - Masquerade combo which will win a two-player game once it fires, no matter how much of a lead your opponent might have. It does this by emptying your opponent's hand and trashing half of it, every turn. Nothing your opponent can do except watch you gradually throw away their entire deck, including all their victory points.
The second edition of the game modified the Masquerade card (if you have no cards in hand, you don't pass or receive any) to neutralize this combo.
The Star Trek Collectible Card Game featured in its first edition the "Telepathic Alien Kidnappers" which allow its player to point at a random card in his opponent's hand and guess the type. If the type is guessed correctly, the card is discarded. Because the rules state that all players draw a new card at the END of their turn, the owner of the kidnappers can guess "personnel" at every guess. Since the player rapidly runs out of cards and usually only has ONE card in their hand, this locks out any new personnel cards from being played as they systematically get discarded. Game over.
The Reindeer Booster Pack in Sopio whose effects include discarding the entire deck, adding all discarded cards to your hand immediately and instantly winning the game for you if played when you are the current leader. This is more or less the entire point of the pack.
WWE Raw Deal had issues with this through its entire lifespan. Some of the more infamous ones:
Andre didn't get a Backlash Deck (think of it as an in-game sideboard a la Magic: The Gathering), but he got a 100 card deck. In a game where you win by depleting your opponent's deck, and the typical deck is 60 cards, this is already a problem.
He begins the game with eight cards in hand and an eight Superstar Value, which means in all but a few matches, he's gonna go first. His ability lets you put a card in your hand on the bottom of your deck, draw one card, then make your opponent randomly discard TWO cards and ends their turn if they either played two cards successfully or willingly ended their turn BEFORE they played a second card, so the opponent's hand is going away quickly.
And then on top of it all, his superstar-specific cards completely break him: "We Face Each Other As God Intended" prevents insta-wins and reduces the opponent's starting hand by one card for EACH Pre-Match card they play. Considering the fact that most decks are dependent on their Pre-Match setting them up for the game, this will either leave the opponent with very few or no cards in hand to begin. And to make things worse, you could use its secondary ability, which causes everyone to discard their hand. Your opponent got to search for one card in their deck but had to reveal it to you. You got to search for TWO cards and didn't have to reveal them, meaning you could grab something to either counter what the opponent searched for or find something to work around whatever they searched for. Other cards were just the frosting on top of the cake of brokenness that is Andre.
The World of Warcraft card game is generally balanced, but in the Throne block (set 6), Horde faction has several cards which are significantly overpowered even for their rarity:
Mazu'kon is a 6-cost ally with 6 ATK and 6 HP in a game where allies with such stat are very, very rare. Then it has Ferocity, bypassing the rule where an ally cannot attack the turn it enters play. Then if it gets killed by conventional means, it can bring out a duplicate of itself immediately.
Daedak the Graveborn is a 4-cost ally which, when killed, will perform Life Drain whose power depends on how many resources his controller has. Players usually will have more resources the longer the game drags on, so the later the game, the deadlier he becomes. Problem is, if he gets killed, he'll shuffle himself into his controller's deck so that he can be drawn again later, so playing him early game is still not a bad idea.
Zodzu, Herald of the Elements is a 5-cost 5 ATK/5HP Shaman ally, which is already pretty good, but if you have another Shaman, he'll proceed to deal 3 damage to any target and heal your hero for 3.
There is also Edwin Van Cleef ally from the 6th set treasure pack. 5ATK/3HP that costs only 4 is already good. But then he has Stealth + Untargettable, which means if he attacks ANYONE in your party, you can only sit and watch. He has no Elusive, so he can be attacked, buuuuut, when he comes to play, he brings two allies with Protector, who can thus shield him from attack (unless the attacker has Stealth). As the icing on the cake, when he does gets killed, you can search your deck for his daughter Vanessa Van Cleef, who'll proceed to smack one of your exhausted (tapped) ally for 4 damage just by showing up. Oh, and the Van Cleefs are neutral-aligned, so both Alliance and Horde can have him.
Plasma Missiles in the 4X board game Eclipse are notoriously Game Breaking. They always fire first in combat and cost no power to fire, meaning that a player can load their ship with nothing but Plasma Missiles and Targeting Computers and blast everything else into atoms before it can fire back. There is a counter - heavily armoured ships - but these ships in turn tend to be weak against everything else, reducing the game to Rock/Paper/Scissors.
A lot of the new secret techs in the expansion, Eclipse: rise of the ancients, directly counter Plasma Missiles. The expansion also add a new game breaker: Antimatter Canon + Antimatter splitter + Point defense.
The Acrobat skill in Descent 1st edition allows the PC to pass through monsters, which can give them access to heavily guarded locations far before they are supposed to be able to enter them. In Road to Legend, an acrobat can completely break the smaller adventure maps by teleporting into the dungeon, rushing through monsters to the save point in the next room, and teleporting out again, all in the same turn.
In the dark card game Lunch Money, the card Humiliation can cancel any card an opponent plays, including another Humiliation. This ability is so powerful that the only winnable strategy is to discard cards as fast as possible hoping to draw a Humiliation before attacking.
Worse, Humiliation allowed an enemy to drop a free, unstoppable attack on you. This included some that could take off from 3-7 life points in a game with only 15 life. Furthermore, you could Humiliate in response to a defense, meaning your original attack hits for damage and any other secondary effects, then you get a free, unstoppable follow-up.
In the short-lived and fairly obscure toy/game "Fistful of Aliens", the order-in figure Jangutz Khan was not only virtually undefeatable in combat, smashing a hole through the Elemental Rock-Paper-Scissors of the lineup, but also permitted you to field a more powerful lineup for the game.
There was once a Mega Man (Classic) card game. In its last set, Mega Man 2's Wood Man was released. Wood Man was extremely broken and could only be stopped by a single card. It's speculated that the deck was what killed the game.
In the Marvel themed deck building game Legendary, Deadpool has a card called "odd ball" which gives him a power bonus equal to the number of previously played cards with odd card values. It itself has an odd card value, meaning that playing 3 Odd Balls gives Deadpool at least a +6 bonus. By stuffing the deck with them a player can devastate villains and almost assure a win; the only solution found so far is to simply ban Deadpool.
That's nothing. In Legendary Villains, Electro has a card ("Supercharge") that gives you +1 attack for every card you discarded before playing it. All of Electro's other cards have a mechanic ("Dodge") that lets you discard them to draw a new card. If you can manage to get rid of enough of your starting cards, and buy nothing but Dodge cards and Supercharge throughout the game, you can end up in an infinite loop, dodging cards and drawing more Dodge cards. You can keep this going for as long as you like, until you decide it's time to play Supercharge and get the actual attack bonus. It does require a lot of time, and specific tailoring of your deck, but effectively gives you infinite attack. The flavor text is "Unlimited power!!", implying that this strategy may, in fact, be the intention of the designers.
Another one is Spider-Man. Nearly all of Spider-Man's cards tell you to reveal the top card of your deck, and draw it if it costs 2 or less. All of Spider-Man's cards cost 2. All your starting cards (obviously) cost 0. So if you buy nothing but Spider-Man cards, it's very easy to end up drawing your entire deck in any given turn. Spider-Man's cards have small bonuses, but when you're drawing your entire deck, you still end up with huge amounts of attack.
A portmanteau term, "CoDzilla", was used to refer to extremely overpowered Cleric and Druid builds through 3E. In addition to incredible magical power on par with the big magic classes like Wizards and Sorcerers (sometimes even better, and unlike Wizards they don't even have to plan their spells out), Co Dzilla got way better combat and utility abilities on top of that. Clerics could wear heavy armour in a total Aversion of Squishy Wizard, plus better base attack progression. Druids were even more broken, as the Wild Shape class feature combined with the Natural Spell feat let them take on incredibly powerful combat forms while retaining their ability to pump out magic. Here's a given example: a Cleric with the Strength domain can boost his Strength by his caster level for one round, plus Righteous Might, Divine Power, Divine Favour and charging with a two-handed weapon, could conceivably kill 99% of the game's monsters in one hit, and this is far from the most optimised Cleric build.
Several Plot Quests in "Lords of Waterdeep" give bonuses that on paper don't sound too impressive, but in the long term gameplay can be broken as anything. There are a few that stand out: Drawing an Intrigue and being able to immediately play it after drawing it if you want to, simply by playing an Intrigue anywhere; being able to refund a single cube used on a quest, including a Mandatory Quest; being able to pick up X resource by simply grabbing a corresponding quest; being able to gain owner benefits on your buildings etc. Some of the standard quests do this too, being able to reclaim three agents at once, being able to draw 4 Intrigues and play them instantly, being able to take all face up quests or all face up buildings at once. The king however, goes to the 'Open Lord' Intrigue, which grants a player immunity to Mandatory Quests and Attack Intrigues, and all for the cost of 'revealing' their lord.
Making these even more broken is when multiple Plot Quests are used in conjunction. Being able to play a new intrigue after playing one is made even better when you score two points every time. Gaining additional resources when you take a certain action is even better when you get owner benefits on a building too.