Exodia in Yu-Gi-Oh!: 7 Trials to Glory. The game has a special area in which there is no banned/limited list, which means you're not only allowed three separate copies of each Exodia piece in your deck, you're allowed three copies of Sangan and Witch of the Black Forest to search them out, AND three copies of Dark Hole, which destroys all monsters on both sides of the field, nuking the opponent's offense while allowing your searchers to do their job. There is also access to 3 copies each of Pot of Greed and Graceful Charity, which all together accounts for fifteen cards with essentially no penalty. The in-game currency reward system has a strong bias towards winning through non-standard means (and Exodia is considered non-standard); what this means is that you get ten times the normal winnings for performing poorly. You can even buy a Deck that is pre-made with all the above cards.
In Yu-Gi-Oh! Reshef of Destruction: The Winged Dragon of Ra's Phoenix Mode can be discarded in the hand, to immediately appear on the field in Battle Mode. note It doesn't necessarily guarantee a win, though, if your opponent can lower its Attack/Defense Points with Effect Cards.
Digimon support are another, since some effect are exclusive to them, and some are ridiculously powerful. Rare types in general have lots of it, who have several number manipulation ability, most notably Toy-Agumon, that make the entire game a breeze.
Yugioh The Sacred Cards had an absurd number of these...so many that the difficulty comes from trying to determine which of these are the most game-breaking. To give a few egregious examples:
Darkness Approaches. Rather ironic, as the real-life card game version is considered one of the worst spells in the game, but there's no denying how utterly game-breaking the SC version is. It turns all monsters on your whole field face-down for no cost at all, allowing them to re-use their instant Flip Effects - a great many of which are clearly meant to be used once per card. One extreme example is Revival Jam, a card that copies itself when activated. Not game-breaking by itself, but with a single Revival Jam and a single Darkness Approaches, you now have four Revival Jams on your side of the field ready to pounce.
Hourglass of Life. It powers up every monster on your side of the field by 500 attack and defense points. Permanently. Not only is this a Game Breaker by itself, but combine it with the above Darkness Approaches and Revival Jam tactic, and you now have two 2500 ATK and two 2000 ATK Revival Jams (not to mention the now 1700 ATK Hourglass itself) in practically no time at all. It is possible to, with the right combination of Hourglass of Life, similar attack boosts, and Darkness Approaches, get a full field of garden-variety monsters to over 4000 ATK, without ever needing any tributes. To call this a One-Hit Kill is an understatement.
The lesson of just how game-breaking these sort of attack boosts are is taught to the player very quickly via Witch's Apprentice, a card the player starts with. It works like Hourglass of Life, except it only powers up Shadow-element monsters. Which in itself is overpowered, because of how the game's Elemental Rock-Paper-Scissors works. In theory, it's Shadow > Light > Fiend > Dreams > Shadow. The problem with this is that Dreams-element monsters are ridiculously rare, to the point where encountering one in the entire game is an event unto itself, making Shadow an unintentional Infinity+1 Element. And on the off-chance that you do encounter such a monster...well, that's what your trap cards are for.
Torrential Tribute, which instantly wipes out every single monster on the opponent's field the instant they try to attack with absolutely anything. And unlike the real-life game, where Mirror Force (a WEAKER version of said card, as it doesn't hit face-downs) is limited to one, you can have three Torrentials in your in-game deck.
For that matter, almost any trap card in the game, because the AI is stupid enough to attack every time it has a chance to win that particular battle. This definition of "chance" includes your face-down monsters. All you need to do is the following: set trap, set monster, end turn, laugh as the opponent kills themselves, attack with all face-up monsters, and repeat until you have won the duel. Traps like Invisible Wire (kills anything under 2000 ATK that attacks you, which no enemy except the final boss can summon without a tribute), Acid Trap Hole (everything under 3000 ATK), and Widespread Ruin (just everything) make the game insultingly easy. To make matters worse, these cards have a deck cost that is absurdly low; deck cost acts sort of like your Character Level, determining the relative power of your deck. Widespread Ruin costs less than the local Goombas to put in your deck, and the other traps cost even less than that. Only Torrential has anything even vaguely resembling a real cost...which you can still pay at the very beginning of the game with minimal effort.
Don't even get started on Coccoon of Evolution. Its duelist level requirement is really quite low. You can have 3 per deck, 2000 defence, needs no tribute. After one turn, it evolves into Great Moth. Another turn later, and that evolves into Perfectly Ultimate Great Moth. 3500 ATK, usually only usable once you've already got two God Cards and quite overpowered.
Ancient Lamp. While its moderately high level requirement means you can't use it until mid-late game without Level Grinding, it more than makes up for this with its ability to completely circumvent the level requirement system. As soon as it hits the field, it can summon La Jinn the Mystical Genie of the Lamp - whose level requirement is extremely high, and justifiably so. La Jinn overpowers every single non-tribute monster that can possibly be summoned by your opponents in the game - and the most common field effect gives it an automatic 540-point attack boost without you even needing to do anything, in a game where the 500-point attack boost from Hourglass of Life is a Game Breaker, as listed above. And to cap it all off, it's Shadow element! Add a Darkness Approaches and a Witch's Apprentice, and with just 3 cards and no tributes you have not one but two monsters with 3340 attack, both of which are nigh-impossible to hit with Elemental Rock-Paper-Scissors, in just two turns. Wish your opponents luck, they'll need it.
The Tag Force games allow you to use certain cards that were only in the anime, like the Dark Synchros. Ordinarily, they're Difficult but Awesome, with their raw power balanced out by the trickiness in using a 0-ATK, high-level Dark Tuner to summon it... unless that Dark Tuner is Doom Submarine. Doom Submarine can revive itself once per game while you control no monsters, and it's Level 9, meaning it can summon pretty much any Dark Synchro you could want. Discard Submarine, Summon Submarine, Normal Summon something else... free Dark Synchro.
Even as Dark Synchros go, Hundred-Eyes Dragon is incredibly strong. It can be played with a simple combo of Doom Submarine/Infernity Randomizer or Mirage, it has a nasty 3000 ATK, and it has three effects. The first shuts down opposing Spells and Traps while it attacks, the second gives it the effects of all the Infernities in your Graveyard (which can include piercing and burn damage, drawing a card every turn, indestructibility by battle or effects, summoning Infernities from the Graveyard, or making it impossible for you to lose the Duel), and the third lets you, whenever the Dragon is somehow destroyed, add any card from your Deck to your hand. This last effect combos perfectly with fellow anime-only card Cursed Prison, letting you Summon it instantly in DEF, so that you'll be able to grab a card when it gets taken down.
Fog Castle, which revives a monster up to four times after it gets destroyed, is incredibly abusable with any card whose effect activates upon destruction. How does knocking 35 cards from your opponent's Deck with a Voltic Bicorn sound? How about using Sangan to instantly draw out Exodia? If that wasn't enough, when Fog Castle runs out of uses, you get to take four monsters from your Graveyard and add them to your hand, massively boosting your advantage in the unlikely event that you haven't just won the Duel.
Philosopher's Stone Sabatiel may be limited to one, but it can be added from your Deck to your hand whenever you lose a Winged Kuriboh, so it's incredibly easy to draw (especially with a Flute of Summoning Kuriboh). What's it do? Let you pay half your LP to add any card from your Deck to your hand. And then it goes back to the Deck, letting you play it again. After three activations, it turns from its original effect to one that multiplies one monster's ATK for a turn by up to 5, potentially giving you a card with over 10,000 ATK... and somehow, that feels like a downgrade.
Trick Battle (a card that reverses the results of a battle so the stronger monster gets destroyed, but damage is still calculated normally) sounds like a quirky bit of fun... until you combo it with Colossal Fighter, a card that can revive itself when destroyed. Summon Colossal Fighter, activate Trick Battle, attack a weaker monster. Your opponent takes damage, but the weaker monster stays and Colossal Fighter gets destroyed. Since it was technically destroyed by battle, you can Summon it back, and since you Summoned it back, you can attack again. Repeat until the opponent dies.
The Destiny Draw mechanic, which automatically placed a designated card on top of your Deck if you were losing, was remarkably abusable, letting you put game-winning cards on top of your Deck in the second turn with some good management of your Life Points. After Glow, which shuffles itself into the Deck and wins the Duel if you draw it on the next turn, was probably the worst offender.
Lost Kingdoms can be broken by ignoring the maximum deck size and instead making a small deck comprised of nothing but the most powerful attack cards you have and "recover used cards" cards.
Online card game Elements: the Game: the Aether element is badly overpowered, with such charming things as multiple completely untouchable monsters and a shield that just makes you invulnerable to non-spell damage for three turns, but the real killer has to be the Entropy element weapon, Discord. If your opponent gets out a Discord in the first few turns, you will almost certainly die, because the special power of Discord is to randomly shunt your quanta around - typically reducing high ones, like, say, the types your deck is based on. The incredibly rare Shards only make balance worse - the aether-aligned one, the Shard of Wisdom, enables you to turbocharge the already obnoxiously strong immortal creatures (which can't be affected by any spells or targeted abilities other than the Shard of Wisdom, making them almost totally impossible to kill) so that they become significantly stronger and deal spell damage, which bypasses nearly all the shields in the game. They also have access to Mindgate, a permanent item which, for an upkeep of 2 Aether quanta, effectively allows them to see what the next card in their opponent's deck will be...by copying it. Finally, they can also spend 7 Aether quanta to activate Parallel Universe, another card which copies any targetable creature on the field. That 10/10 Golden Dragon you brought out? They now have one too, and at 5 less total quanta cost.
The Savants, which are rather ok statwise (as in, easy to kill by a majority of other creatures and spells in the game), are strong because as they level up, they encourage you to play weaker cards for them to trigger their effects. Because you can't level up all your cards so soon, having a few high-levelled Savants means you have a lot of low level cards yet to be levelled up, and because playing those low level cards now reward you with levelling them up AND triggering the Savants.... well, suffice to say, Savants are powerful enough to get banned in some tournaments.
Zimus, the Undying. Once you can level him up to the maximum, there are very few cards that can answer him (only reliable way is to use Botanimate, a Baleful Polymorph spell). Why? Because every time you kill him, he'll just come back from the dead. IMMEDIATELY.
Everflame Phoenix. This card's level 1 stat is already pretty amazing, but its level 2 is epically horrible. That is, until you gain a rank (which happens once every four turns) and that level 2 instantly transform into a level 3 monstrosity with high enough stat to one-shot 99% of all creatures available in the game, and when it dies, it will nuke the creature face-to-face with it, along with their owner. Then it returns as a lowly level 2 Phoenix. Which can later grow into level 3 again.