I'm not very into video games and actually I've grew a little sick of them, but I like rpg and adventure games.
Those ones where you have to overcome obstacles to accomplish a mission, such as Tomb Raider for example. I also love Mario games, specially Mario RPG, Mario Kart and Mario Party; Zelda and Kingdom Hearts.
Guitar Hero can also be so addictive!
Also, Online RPGs are cool. It's much more fun when you can play in groups.
Not so much the leveling up, but creating a background history for your character and role playing... immersing into the fictional world...