I like some games, but I don't know that I "enjoy" games so much as view them as mental exercises. I don't "play" games. I solve them. As a result, the notion of "playing games for fun" doesn't appeal to me. If I can't solve (master) the patterns, data, or trivia behind the game, it isn't going to be that interesting.
Not winning a trivia game? It means I need to read and learn more.
Not winning at chess? Time to read and practice more until I get better at the theories of various masters.
I don't like random games, either. Too many "fun" games are based solely on luck. I'm not interested in luck. You can't master and win luck.
With the desire to solve and win, I'm not the least bit "fun" when playing games.
Over the years, I have tried to develop my own games. These were not random things. I mapped out alternatives to chess and checkers when I was in elementary school. Movements were dictated by the square of the board, with pieces all the same. The idea was to create logical patterns that could be mastered.
I enjoy darts. Again, it is about mastery. Skill matters, but I don't have to be a great athlete to participate. You can easily try to best yourself, constantly, until you are capable of near-perfect aim.
The perfect game is one I can master alone, or against a computer. My mind against the patterns.
We own a Nintendo DS and the games we have purchased: crossword puzzles, brain games, word games, and chess. Not a single "video game" to my knowledge. I do like PacMan, but that's a puzzle with a pattern. I am not interested in violent games or randomness.
People don't enjoy playing games with me because I am unlikely to be enjoying the social aspects of the game — I'm entirely focused on beating the game's design.