If you're not going to run any Horizontal games in it, then 2 buttons will do for you. Use SortInfo, then delete all non-vertical games, then delete all non 4way joy games. Sort the remaining list by number of buttons. There are only 4 games that use more than 2 buttons. Tutankham uses 3, Chinese Hero, KO Punch, and Street Heat use 4. There is one other game that might make a difference, Toobin'. No joystick, but it needs 5 buttons.
If you up it to an 8 way, then things change a good bit.