One Button. Five Games.
Before I list them, let me define a one-button game quickly. One button means a single point of interaction that the player can press and depress. Anything more than that is no longer one-button; games that use a mouse are not one button (X/Y axis movement is more complicated than pressing / de-pressing), games that use directional pads are not one button (up/down/left/right are each a button), games with analog sticks are not one button (joystick-only is a single point of interaction, but the actions are more complicated than simply pressing / de-pressing). Also, I'm only concerned with the core gameplay, so menu navigation or name entering does not disqualify a game.
Nintendo's Rhythm Heaven requires one button (represented on the touch screen) to be pressed in all of its 50 game sections, though it has more buttons to get through menus. In the game the top screen visualizes the theme of each game, such as putting together bolts or singing to a crowd. Each game has rules on when and how the player should act. The three basic actions that involve the single button is a quick tap, a held press, and flicking the button away. Some games are simple, like the bolt game where the player flicks whenever two bolts align, while others are more complex, like the singing game, where the player listens for song phrases and must tap, hold or flick when certain words come up. Again, cramming a ton of functionality into a very simple interface.