A sample application demonstrating how to integrate a LibGDX Android game with Google Play Game Services (leaderboards and achievements).
The game itself is one of the example applications included with LibGDX (Super Jumper) https://github.com/libgdx/libgdx/tree/master/demos/superjumper
Open source and available on GitHub - https://github.com/TheInvader360
Companion tutorial on the blog - http://theinvader360.blogspot.co.uk/2013/10/google-play-game-services-tutorial-example.html
Play a more complete version of the game (Super Jump a Doodle): https://play.google.com/store/apps/details?id=com.theinvader360.jumping.superjumpadoodle.free