Super Mario Bros or SMB3. Like come on, come on. And Tetris everyone should also give a chance.
Doom, if you've played that you really have the jist of where FPS games are about.
Zelda OoT, I think that was the first game I played that was truly amazing. Games before that were fun, very well done, etc but OoT changed what I thought games could be. It brought together good story, fun gameplay and puzzles, and an interesting world to explore. Probably also one of the first times a game had real immersion for me too.
With those you get a good sample of addicting simple stuff, high action fun, and stuff that's deeper you can really get absorbed into.