1) For some reason (potions accident, curse, maybe he's part Veela or something) Harry goes into heat and it's up to Snape to keep him from committing any indiscretions--at least, keep him from committing any indiscretions with anyone else.
2) A courtship fic where Harry has to court Snape, reluctantly at first, then becomes more determined to win the man over when he realizes he has competition.
3) As a baby, Harry is sent to the future (maybe for his protection), where he is discovered by his older self, who enlists Snape's help taking care of him.
4) AU Harry is a street urchin and Snape catches him trying to pick his pocket.
5) Snape places an order for a rare potions ingredient and ends up with a crate full of Harry Potter instead.