Chessmaster1000: All pieces on the ace point except for two men each. Black has the 15th piece on the Bar and White has the 15th piece on the 5-point. Both sides roll 5-5 ad infinitum.

By "all possible, different Backgammon games" do you mean the variants? In which case we need a list so that we're talking about the same thing. From VogClub I know a few variants:
Tapa, Narde (Feuga) and Crazy Narde (Gul Bara) are finite as there is no sending back to the bar.
Longammon, Nackgammon and Acey-Deucey are the same as Backgammon.
Hypergammon is, of course, ripe with infinity.