Technically you only need two, but optimally you'd use three. Double that if you put buttons on both sides of the joystick.
Only two games on the 60-in-1 use three buttons - Breakout and Gunsmoke. Breakout only uses them to decided the block pattern at the beginning of the game and most people doin't even know that exists so will never miss the 3rd button. Gunsmoke has an option in the setting to play it with 1 button (you fire in whatever direction you are facing) but a true Gunsmoke fan (like me) hates this and would prefer having the three buttons...