Hmm i understand. But you must have a mistake in your previous post. Please correct it......

You wrote:
The number of different backgammon games is finite if, and only if, there's at least one game with a position that repeats itself.

I guess you should replace finite with infinite. Right........?

And of cource an easy proof that these special positions exist, is to choose M=N+1 and have both opponents at the bar and choose such a dice roll that doesn't get any of the 2 from the bar, for 2 consecutive rolls........

But that really proves that the number of different Backgammon games are infinite.....? At a first glance it does, as k can go to infinity but perhaps this is not critical......
I will think about it and answer later......