1 standard 16A group can deliver 3600W max. (in 230VAC countries !!!)
A regular upright video game is about 100W maybe a bit more. My SW cockpit is rated at 220W but I think it's a bit lower in reality.
They always put "worst case" numbers on devices. You do have to realize that during power-up, the amps are a lot higher than when running normally. This could create a surge if you power up several with one switch.
Anyway, 3600W would mean about 36 uprights MAX. I wouldn't go over 30 to stay on the safe side.
At the early Eurocades in Ridderkerk we had all machines on 3 groups, carefully decided (about) equally. We never had a problem (apart from when RobinHolland was fiddling with a cab

).
Anyway, always better to be on the safe side, extending later is always more expensive that including it now.