There is an "impossible" combination of joystick outputs that I'm surprised nobody ever made use of, namely up+down+left+right. In fact any combination that involves two opposite directions together wouldn't usually be possible (7 in total). Someone clever could have used this to give us a start/select button and maybe some others, without the need for extra pins or multiplexing.
Also one thing I have in mind now is an adapter for a playstation pad that connects to both the controller ports, so the analogue stick can be used as the mouse.