Card games like spades,hearts or Gin Rummy have solid mobile versions and decent privacy.For classic arcade vibes,Tetris and pac-man still have active global leaderboards.

Man I was really hoping this was reminiscing about classic ipod games.

Anyways since no one is listing real games, I'll list 2 I enjoyed a lot.

- i dig it - zombie farm

If you're wanting to actually improve over time, you'll have to drop a few. There's not enough time in the day to be a master of all trades. I'd suggest spending more time on Chess - Do a small learning component where you read a book or do some opening theory, then do a portion doing puzzles to learn tactics and spotting common positions in Chess games, then finally play some 10 minute games to put into action what you have practiced.