I don't think you could ever get this working the way you want, without modifying the game software itself. The closest you could possibly come is making a circuit that blocks pressing of the "start" key more times than the number of coins that have been inserted. I don't know how that would behave with things like continues, pausing the game, accesing a menu, etc. etc.
It would go like this, shown in a table with coin-insert events, start-press events, and START indicating whether the game sees the start button or not. Counter counts up when a coin goes in, counts down when start is pressed, and only allows a start press to pass if the counter is not zero.
counter | 0 1 0 0 1 2 3 2 1 2 1 0 0 0
ins_coin | * * * * *
press_start | * * * * * * * *
START | * * * * *
Hope that comes out looking anything close to a table. It probably won't.
By the way, the circuit to do this would be worthy of a final project in some electrical engineering classes. So if when I say "7402" you automatically think "NOR gate" and know from memory which pin is VCC, you can probably build this.