Simple solution, which would also cover you against 1.d4, 1.c4, etc, is to play the Modern Defense. It's not exactly the cream of the crop at the GM Level. Most players below 2000 that don't play the Modern themselves end up clueless when facing it.
I need an exciting off-beat opening vs 1. e4
I used to be a columnist at Chessville and my column was called
"The Search for Dragons and Mythical Chess Openings". In that
column I did exploration of unknown chess gambits and chess
openings. You might be interested in the Panteldakis Counter
Gambit against the Kings Gambit. It has the moves 1.e4 e5
2.f4 f5. There are other chess openings that you may be
interested in. See link below:
Chessville Web Site
The web site Chessville is no longer operational. My column has been placed at an archive site. Recently I have been on the archive chess site to recover my articles and game files. See link below:
The Search for Dragons and Mythical Chess Openings

You can play the Alekhine 1 e4 Nf6. It's definitely exciting and forceful. I hope it's offbeat enough to you.
I will consider it but I never really thought much of an opening in which you move your knight 3 times in 4 moves, personally. It just seems to break down what I know about the rules of the opening
I think Alekhine's defense is a great suggestion for an offbeat reply to 1.e4. It's totally sound (almost certainly more so than the Scandinavian) and seems to lead to fairly unique positions that many White players would not be prepared for.

I used to be a columnist at Chessville and my column was called
"The Search for Dragons and Mythical Chess Openings". In that
column I did exploration of unknown chess gambits and chess
openings. You might be interested in the Panteldakis Counter
Gambit against the Kings Gambit. It has the moves 1.e4 e5
2.f4 f5. There are other chess openings that you may be
interested in. See link below:
Chessville Web Site
The web site Chessville is no longer operational. My column has been placed at an archive site. Recently I have been on the archive chess site to recover my articles and game files. See link below:
The Search for Dragons and Mythical Chess Openings
Thanks for the replies, really interesting stuff. g5 certainly seems to catch your opponents off guard, that's for sure.

I agree that you should look into the Alekhine Defense. It's more exciting than you might think and will challenge your conceptions about the rules of chess.
Balogh's.
1.e4 d6 2.d4 f5!?
It isn't great, but it's playable, and not being "great" isn't going to bother anyone who plays the BDG anyway.