There was one that I strongly tried to steer away from the insane, and one that I wish I would've disallowed.
The one I took a strong hand in, shall we say, refining was for a Vampire game, from a girl that hadn't roleplayed before. She wanted to be, basically, everything. And the most important person in the game. Had her concept been carried through, she would have been an Indian Princess (so not kidding) who had a graduate degree and became an assassin with knife expertise under the guidance of her radical professor from college, who under her cover identity was traveling with circus freaks, and her boyfriend was a rock musician, and then she was embraced to become a vampire because a Prince's rival thought she was ideally suited to kill the Prince so that he could assume power, but the Prince found out and killed her sire, but left her alive because she was so useful.
I blinked. I said, "Is this character by any chance named Mary Sue?" She didn't get the reference. She said that her name was Sapphire. Huh. An Indian Princess assassin knife expert with an MBA that lives with circus freaks and has a rock star boyfriend, named Sapphire. At this point, I was seriously considering running out of the house screaming.
Let's just say I intervened and heavily edited that backstory. We never got to play because of other circumstances with that group, for which I'm almost thankful.
Another one that I wish I had the foresight to disallow was an alternate reality Multiple Man in a heroes game, who was an Elvis impersonator. Strike that, he was in truth all the Elvis impersonators. They were really all just one guy, a mutant. It was funny, and so I allowed it. Then the player decided the best way to roleplay Multiple Elvis was to try to challenge everyone to a fight every chance he got, and when that failed go off in the middle of fights to make peanut butter and banana sandwiches.
I've been more fortunate with D&D and Star Wars games.