If you haven’t played Canabalt, you should give it a try. This is one of the games to define the single button endless runner type games. the original was made in Flash. There’s also an IOS and Android version. I hear the Android version is better with different worlds and things.
Essentially the gist of the genre is that you are running forward avoiding obstacles. things get tougher as the game progresses. There’s lots of variation. I’m linking to an article here by the author of Canabalt talking about what went into making Canabalt and the decisions for different screen sizes, and game mechanics. Very informative if you wanted to make a similar game.