It can really depend. I get more lasting satisfaction from deep games, but they can be hard to get into, and that's a turn off when I'm looking for some instant gratification. I need a few hours available for Heavy Rain, but only 30 minutes to enjoy a shooter.
It also matters how the game is presented. Stories are often there, but you have no idea with sand box games. I'll be playing Red Faction: Guerrilla, blowing up some buildings, and suddenly ask why I am doing it. What's the point beyond running up on a hill and setting off my charges to watch the building collapse in a glorious series of explosions.
Then others have the story hidden in large boxes of text in some journal. Why am I doing this? Hold on, let me check the glossary. Oh, yeah, that's why. Not so fun.