AbigailII: You got it wrong.
Correct! I know.....I felt that my example was wrong, but i never really believed it was. Seems stupid right? It was just a matter of not thinking about it a bit more........

I will add later something to the very interesting point of: "Say there are a finite number of games, call this number M. But that could not have included a game that reached the position I described above and then continued with M rolls of 1-1 on both sides, followed by a roll of 6-6. Ergo, there's no limit on the number of different games.

Wil: For every move to the infinitum, the probability is 1/36, why would it be smaller at some point?
For every move it's 1/36.
For 2 moves to happen is (1/36)^2
For 3 moves to happen is (1/36)^3
For an infinite number of times it's zero.

If we count all the possible ends when one player doesn't throw 5+5, we get an infinite amount of games.

How do you conclude that.......?!?!?!?