I was just introduced to the Royal Game of Ur, a c. 3000 BCE possible predecessor to backgammon. You can watch a comprehensive intro to the rules and see a game being played here:

Trailer here:

You can find with a short google various versions of the rules and printable game boards/game pieces.

There definitely are some elements that may be an improvement over backgammon, such as the ability to roll 0 and it being much less studied/solved.

There's no definitive betting procedure for this game. I'm not even sure people currently play this game for any significant amount of money, but it seems ripe for the playing. Has anyone in here played this game for money?