There are no games where players 3 and 4 need 6 buttons each.
Probably the closest is D&D shadow over Mystara (capcom 1997), 4 players with 4 buttons per player + 1 optional button per player used to show the inventory + 1 start button per player.
Normally the "start" button and the "inventory" button are the same one.
Honorable mention D&D tower of doom (capcom 1995?), 4 players with 4 buttons per player + 1 start button per player
They are both cool games tho
Anyone know how many buttons Golden Ax 2 is? 4 players with 3 buttons per player + start I think? after SOM it's probably the best beat em up