Personally, I don't play a lot of games. But Super Mario 64, Tetris, World of Warcraft, Super Mario Galaxy 1 and 2, the three Paper Mario's, Super Mario RPG, were my games before starting Donkey Kong.
I don't really play any other arcade game other than Donkey Kong, I might try to get good at Tetris (because there's a lot of competition, and that's what I like!).
If I really wanted a "competitive" game, I would play Starcraft 2... but only Koreans, I can't do 10 actions per second (mine is pretty much 10 actions.. per game, haha. /close enough).
Mike: League of Legends could be an interesting game to play. It's not a fast game like Starcraft 2, you don't need to play 12 hours per day like WoW to have the best gears, etc. And it's a very popular/competitive game (extremely popular on twitch.tv). Maybe that could be a good game to try!