Bxe6
1.e4 f5 : me and my oversized ego vs all of you
engines also all agree with Irontiger's moves.
We're only 12 moves into the game at this this point. While 1.e4 f5 is not a very common opening, it has been played, and there are opening databases and opening books that have games with this opening. It's very possible that IronTiger is using something like this. IronTiger never said he wouldn't use opening databases, he said he wouldn't use engines.
It's also possible that IronTiger is simply following good opening principles, has a lot of knowledge of this opening, or is simply a very strong player.

We don't have our friendly Panda and Houdini (or our acting captain Doggy), but Bxe6 looks like a consensus - any objections?
Houdini also likes Bxe6, at least up to 28 ply, when I accidentally disconnected the engine, but I doubt it would change its mind.

engines also all agree with Irontiger's moves.
Considering they don't always agree with each other, that does not sound credible.
Please feel free to post any engine's analysis to back up your claim, though. Make sure to have screenshots, or people might be lured into the irrational thinking that you are just a troll.
As for the means I am using : no computer, no database, I did not even use the analysis board up to this point. I have spent about a combined 30 minutes of thinking up to this point, which is probably longer than you should spend for 12 moves in a serious game.
I am guessing 2 or 3 moves are not top-3 computer up to this point. Moves 2, 3, 9, 10 and 12 are certainly in top 3, and the rest cannot be so bad either.

Off-topic : for anyone who might enjoy it, similar threads have been started : Polish defense (1.e4 b5) by Talonflame_Fan and Muzio gambit (some king's gambit crazy line) by Dude_3.
On topic : ...Bxe6 seems to be the move (to be honest, I expected it), so Nbd2 :

GNU suggests Qf7 - this is where computers loose me - I don't see why. Maybe to free the Knight or a future Bishop move giving discovered attack on the Queen?
Qf7 with intention to play Qh5, and you have a strong threat with queen, knight and bishops. Also, why playing h3 and disturb the pawn structure when you can capture with the queen? It's also a good waiting move.

Following LoekBergman's example, my post is in white:
Houdini 4 pro x64: 13...h6
Komodo TCEC: 13...Qf7 very very close was ...h6
Stockfish DD: 13...Qf7
All of them at 30 plies. If needed I can try with further plies.
In my opinion, this is how they beat Houdini, their evaluation criteria is better. Houdini sees the move, but fails to choose it.
It happens to us as well; we see candidate moves, but fail to choose the best one, whereas a GM chooses the best one.

GNU suggests Qf7 - this is where computers loose me - I don't see why. Maybe to free the Knight or a future Bishop move giving discovered attack on the Queen?
Qf7 with intention to [removed].
You need to add a preamble to this comment or otherwise make it spoiler-proof; making statements right out of the gate like this makes it impossible for Irontiger to avoid reading them.
Maybe it's time to take back the pawn now. Bxe6
Komodo agrees.
I think all engines will agree with you.