Well the asnwer to the theoretical question of what is the shortest game possible in Backgammon can't be answered, since the question is not well defined.
It is not, because we have to say if a "pass" move is considered as a move or not, we have to exclude resignation (of course!) of a player, we must care only for single point games and not include doubling cube(of course again!).

So if we consider the "pass" move as a move for a player, and not take into consideration the resignation and doubling cube options, then the shortest possible game WHICH IS ALSO LOGICAL (meaning it doesn't contain bad moves) must*** be the following 17 moves@@@ game:

*** I say must, because it hasn't been proved.
@@@ A move is defined like the move each played does, and not like a sequence of 2 moves, that of one player and that of the other.

1) 43: 13/9 13/10
55: 13/8 13/8 8/3 8/3
2) 55: 13/8 10/5 8/3 8/3
55: 13/8 13/8 13/8 8/3
3) 66: 24/18 24/18 13/7 13/7
33: 24/21 24/21 8/5 8/5
4) 66: 18/12 18/12 12/6 9/3
66: 21/15 21/15 15/9 15/9
5) 66: 12/6 8/2 8/2 7/1
66: 9/3 9/3 8/2 8/2
6) 66: 7/1 6/off 6/off 6/off
66: 8/2 6/off 6/off 6/off
7) 66: 6/off 6/off 6/off 6/off
66: 6/off 6/off 5/off 5/off
8) 66: 5/off 3/off 3/off 3/off
66: 3/off 3/off 3/off 3/off
9) 66: 2/off 2/off 1/off 1/off

(The above game was given/invented by Kit Woolsey i think.)

Many more 17-move games can be found with the above restrictions(With many bad moves though). I have once read that the shortest game possible with the above restrictions is 16 moves but i haven't seen the game and also many disagree with this claim. I don't know. Does anyone else knows?