Hi all. Our team had an idea that we decided not to go with, but I wanted to share in case anyone else might benefit from this.
The premise is that you win by button mashing fast, but only if there is sufficient variation in your buttons.
When button mashing, our inclination is to press buttons in a pattern (eg roll your thumb across the D-pad or face buttons). This game would detect that and punish you for it. You succeed when your button presses are truly random over a period of time.
This concept subverts the idea of what a game is, and is a different way of interacting with the controller. Ideally it would make the player say "what the hell?" and freeze up, because they really want to press the buttons in a pattern. This is a confusing, grueling challenge, that would be extremely difficult for a player to strategize around. The goal is to make the player feel totally powerless and like this is a challenge that can't be overcome. A two player mode could pit players head-to-head.
Our team decided that this sort of game is more like a form of postmodern art than an actual game, so we decided not to do it. Please feel free to use this idea for your own game.