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If helpful, there are some useful examples here: OneSwitch PC Games - especially so the cause and effect and low pressure area. Not all would fit within the parameters of this competition, but it may give some examples.

I've even seen a couple of one-button (and you only need to touch it once) games in the arcade. Both redemption/ticket games. One based on Frogger, where you press an illuminated green button, and it sets a frog off (up - pause - up - pause - [repeat]) - across a busy road, then river, trying to get to a safe lily pad.

Another was a bit like Angry Birds, with a human cannon ball, where the cannon constantly automatically adjusted the angle. When you hit the switch you fired the man out across a city scape, trying to get him to land on top of a sky-scraper. Basic, but shows just how far you can reduce controls to make a game.

Good luck, and I hope it doesn't put you off.