Sassoongammon is a fun variation where the player rolls 3 die and the opponent chooses which 2 of the 3 will be used. I recommend some variations where you stop doing this when there is no contact and just play with 2 die from that point on.
I really like Backgammon to Lose a lot. For the first three turns for both sides, you play as BAD as you can. Then you change sides (turn the board around) and play as well as you can from there. You will find yourself in extremely unusual and interesting positions and will have to be very creative to win.
Several tournaments have tried my "Simborg Rule" in side events. The rule states that you cannot make a point on the opening roll, or if you roll 6-5 on the opening roll, you cannot run one checker. This takes away the huge advantage gained by some opening rolls and causes the game to be more complicated from the start.
Nackgammon is also a fantastic game (start with 4 checkers back instead of two, with checkers on the 23 and 24 points) as this causes the game to be very complicated.
These variations are not only fun, they require the players to be creative and play from positions and situations that are unfamiliar.