1. Minecraft
The classic open world sandbox game which made its debut on PC many years ago was more recently ported to consoles and mobile devices. The mobile version has come a long way and gained many new features, some unavailable on the PC version!
You set your own goals in the game and it is very flexible time-wise, save whenever you want and pick it up just like you left it. The definition of a good mobile game.
2. Tiny Wings
A forgotten gem that dominated the App Store leaderboards years ago. Recently updated with new levels, worth a visit if you have never played it.
The pastel art scheme and whimsical nature of the game are very relaxing and offer a replayable experience. The goal of the game is to make it as far as you can in the traditional time run mode, or to compete against computer controlled birds to get to the nest first and eat the biggest worm.
3. Death Rally
A more violent combat experience than traditional racing games, the player views the racetrack from a top down perspective. It is up to the player whether he wishes to race peacefully or make use of the array of arms available to him/her to rain death upon opponents and race to first place.
Not as appealing to the general public as the previous two, but worth a spin if you enjoy racing games at all.