Is a wasted tempo with an early ...a6 or ...h6 by Black in the opening really compromising? Is a wasted tempo in the opening for Black with an early ...a6 or ...h6 really dangerous? For example, in the following opening:

[FEN ""]

1.e4 e5 2.Nf3 Nc6 3.Bc4 a6 (or 3...h6)


Or in the following:

[FEN ""]

1.d4 d5 2.c4 e6 3.Nc3 a6 (or 3...h6)


Or in the English Opening: 

[FEN ""]

1.c4 a6 (or 1...h6) 


And so on. Is Black's position really compromised, or can he afford to do that without much trouble?
At such an early stage, there is plenty of room for speculation, but the answer is that it can be dangerous in some positions, and not so dangerous in others. I will go into detail in a moment. The biggest drawback of playing those moves so early is that you are not yet sure if they will fit into, and if they will be useful (very important), as the position develops. A good rule of thumb is to play moves that you know you will need earlier, and then see what other moves fit in with your plan as the game has developed.

In the first position, you have a double-king-pawn opening, aka an open game. It is called an open game for a reason...because the position will be very open. Very open positions often rely on tactics and tempo, so losing a tempo on move three could easily evolve into a lost game by move 15. I would say in that position, it is the most dangerous.

In the second and third positions, both are closed games, and thus, losing a tempo is not as likely to be fatal. Again, whether you can make a6 valuable in those positions, it is not clear. In either case, even if it does not lead to an immediate win, I would expect white to have a MUCH better chance at obtaining a solid +/- out of the opening, instead of the more typical +/= or =.