I think a good rule is 4-5 machines per 20 amp breaker. This allows for a good cushion for those times when the machines have everything going on at once. Also, remember that any amp draw will vary depending on the number of LEDs you have in your machines.
I have 5 20 amp circuits in my game room. The most I have on a single one is 3 pinball machines and 2 video arcade games. I also have each circuit on a switch.