There are quite a few that come to mind, but none so much as the titular last boss from the Mega Drive fighting game Eternal Champions, who is always my go to answer in threads like these. How is he broken? Let me count the ways:
1. He has all the staples of a coin-munching fighting game last boss. Insane speed, crazy damage output, broken priority, moves that combine all the best bits of other characters' movesets...
2. The AI reads your inputs like a fiend (the developers later came out and admitted this), meaning that any move that you throw at him in a 50/50 situation will be beaten. The only way to hit him (and indeed every enemy in the arcade mode) is to take advantage of gaps in his attacks, or abuse glitches in his AI and the game mechanics.
3. He has five different forms, each with their own full life bar and move set you must learn. You get one life bar (albeit with a small health boost between each form) to beat them all. Lose to one? Unlucky. Back to the start of them all.
4. His first form has a move that makes him temporarily invincible. There are loads of cheap moves like that in the game, so it's not that unusual, but they're regulated by the game's inner strength meter, and the cheapest ones cripple your special move output for a long time after usage. The difference? The Eternal Champion has infinite inner strength. Oops. In short, if the AI decides you are going to lose, you will lose. There is no way around it. Block and he'll throw you or chip off your health with insane chip damage. Try and fight back and your moves pass straight through him, meaning he'll just kick chunks out of you.
5. One of his other forms has a move that puts an aura on him that will hurt you whenever you hit him, and makes his normal attacks chip your health when you block. Ditto to the above, basically, in that if the AI decides to use this move enough you will lose, with the very slight advantage that you can beat him by just praying that your health bar lasts longer than his does. Of course that cripples your health bar for the next form. Better hope that he picks this one last.
6. You have two rounds against him, and you have to win one. If you lose both rounds, it's game over. No continues, no try agains, no nothing. You get a bad ending, and you're kicked back to the title screen to play through the whole (not exactly easy) arcade mode again. Two rounds. So two chances to become acquainted with five different movesets, to find holes in said movesets that are sufficient enough to beat the psychic AI, and to pray that he doesn't decide to spam one of two moves that are pretty much an instant loss for you. No, this is not a boss you're going to beat on your first try. Nor your second. Nor your third...
TL ; DR, he's an SNK fighting game boss if you took him out of the well-polished, fairly well-balanced fighting engine of an SNK game and stuck him into a below par fighting game with some extremely ill advised design decisions. Which is to say pretty much unbeatable by conventional means.
I finished the game with every character as a kid because kids are insane like that, but only by abusing a glitch that allows you to throw an opposing character indefinitely when they're trapped in a corner. And even then you have to pray that you can get him into a corner without being killed first, and that you can continue timing your throws correctly to take off five of his health bars so that he doesn't break free and end you.