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1...a6. Miles defeeated Karpov using 1...a6
Modern 1...g6. Play hippo moves after that if you want off-beat
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.
1. E4 Nc6 2. Nf3 f5... this is the Colorado Gambit. It might look a bit strange but there are so many ways for White to lose very very fast. There is an amazing Video on youtube about this. Just search there for Colorado Gambit and you'll learn an very effective surprising Weapon against E4.
Seeing all the BS that is written before me (the Alekhine or Scandinavian are not theoretical, ha ha), here is mine :
1.e4 f5 is quite offbeat.
Exciting, I don't know. Does the prospect of being checkmated before move 10 excite you ?