Counterattack and funny games, and space for creativity. What do you want more?
What does the Alekhine defense possible have for black???

I believe the basic idea is that it lures whites pawns further forward than he might initially like. This can put a little pressure on white to construct a large pawn centre (e.g. the four pawns attack) and this can be exploited by black to wreck the pawn centre (e.g. by c7-c5 or f7-f6) and take advantage. Clearly if white is on top of this black may have to weather a storm. I don't think the theory is entirely straightforward so it can lead to interesting games. I'm sure others can provide more concrete examples.
The alekhine defense is terrible. What does it possibly give black over white?