What's Black's Sharpest Response against 1.e4?

It doesn't have to by healthy, just sharp....

Alekhine or sicillian

He said it didn't have to be healthy ;).

Probably Latvian gambit or something.

Hmmmm, I'll have to try it out...

ha! i've got something perfect for you: 

SCANDINAVIAN OR FRENCH!!!

The Latvian Counter-Gambit and Wilkes-Barre Var of the Two Kts Defense are both very sharp but in turn-based chess you're going to really take your chances on not being "outbooked" instead of outplayed. I'd save the super-sharp openings that are either considered unsound and/or rarely played by GMs for thematic tnmts where everyone has to play them, otherwise they usually turn out to be a form of odds giving as Nimzovitch once remarked after playing 1.e4 e6 2.d4 d5 3.Nc3 Nc6 with the Black pieces. 

And of course almost any opening can turn out to be extremely sharp depending on the variation employed, such as the Poison Pawn Variations in the FD and SD; but a lot of sharp openings only appear OTB because BOTH players agree to employ them.

1... f5

Sharpest is Alekhines Defense as it immediately questions white's e4 advance. Sicilian with c5 can lead to razor-sharp lines, too.

 fireballz, why not 7.Qxf5 Ke7 8.Qf7# ?

 even better!:D thx for that one, ya!?

So many Alekhine's answers!  Alekhine's is very, very positional.  Terrible answer to the question.

My vote for sharpest would be the Philidor counter-gambit.  1.e4e5 2.Nf3d6 3.d4f5!?  It tends to be a lot more tactical and decisive than even the Latvian.

Scandinavian for me :) 

Analysing and going through some Sergei Tiviakov games will definetely help you alot in Scandinavian!

Good luck on whatever response you choose though.

it you want the sharpest variation that is regularly employed by grandmasters today and surprisingly one of chess's most popular i would recomend the poisoned pawn nadjorf.

.................

Unless you're playing the Nf6 lines in the Scandinavian, the goal is to get a slightly less ambitious Classical Caro-Kann, with the side benefit of having avoided the Panov, advance, etc.  The Scandi, at its heart, is designed to be a very solid defense, and to avoid sharpness.

1..Resigns! is my favorite opening