My favorite game is still Solitaire. This may tell you I am old. But Solitaire is a non-violent game and it challenges the mind. Chess also challenges the mind but there is violence, the pieces eating other pieces. The new games in the computer or mobile phones are just not attractive to me.
Chessmaster and the NBA Live series have been my top favorites ever since I started playing on the computer. Reflects my 2 most favorite real life games - basketball and chess.
My favourite games include Grand Theft Auto and the FIFA game series, especially on Xbox 360. I remember when I first got my PC though, I used to love the tycoon and entrepuener kind of games. I loved them.
all rpg games like diablo 3 and all grand theft auto games. Also command and conquer red alert 1 and 2