Joust — Players face off by revealing the numerator of a rational function then make choices for the denominator based on the revealed opponent numerator. The player with the higher horizontal asymptote wins, just like in Joust.
Battleship — Similar to traditional battleship rules but students fire numbers at each other in addition a location on the grid. Instead of being worth 3-5 pegs the battleships are represented by a rational function. Your ship is sunk if your opponent hits it with a number outside of the functions domain. (Dividing by zero sucks the ship into a blackhole, square roots of negatives make the ship imaginary, haha) You could make the grid smaller so the chance of choosing a coordinate with a grid is higher, and the risk comes from shooting the right number to sink the battleship. You could tweak the rules so that students either have to report a “complete miss” (no ship) and a “no effect” (ship but the number was in the domain of the representative rational function)
I think these would work great as review games for rational functions.