Double Choice Backgammon - each player has 2 sets of dice.
Triple Choice Backgammon - each player has 3 sets of dice.

Below - I'm explaining the triple choice.
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At the start of the game, each player has 3 sets of dice.

Each player rolls their 3 pairs, giving them 3 moves to choose from (where the other opponent can see their 3 choices)

example:

Player 1 has: 6-5, 3-3, and 5-1
Player 2 has: 6-2, 5-5, and 5-2

To start, the person with the "biggest" pair wins. (Player 1 has 6-5, so goes first. If there is a tie, it goes down the the second "biggest" pair.. and then third, and if all 3 match - then re-roll everything)

First player can choose which one of the pairs of dice to use for their move. So for example, player 1 decides to use the 3-3, and makes their move. They then re-roll ONLY the dice that were used (leaving the 2 unused there), and then their turn is over.

... and so on and so on.

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Other rules:
If possible, you have to pick a pair that will allow you to use BOTH dice. (If one of the dice has a 6 and you can move 6, then you can't pick that pair if another pair allows you to use both dice)

If you can only use 1 dice from all rolls, you have to use the higher of the 2 numbers.

If you can't move with any of your dice, you can pick 1 pair to re-roll. After the re-roll, your turn is over and you can then use that re-roll on your next turn.

Most everything else would be similar to backgammon. What this game adds is a lot more strategy - since not only do you have multiple things to choose to do each turn, you can also see all possible moves your opponent has to make.