In George R. R. Martin's shared super hero world there was a character like that. Only he so feared what his power might be (and the repercussions of it) so much that he fought sleep as long as he possibly could, to the point of insanity.
You might want to check it out. The series is called Wildcards, there are Aces and Jokers. Aces are the people blessed with usable superpowers while Jokers are those whose mutation is an hindrance. (The card allusion being the randomness of the card you might draw from a deck.)
It's an interesting premise.
Bob Burden said:
Pie for the pirates,
wine for the dogs.
Yeah, basically it's some guy who wakes up every morning with a different superpower/mutation/whatever. 
