Haha, oh I love this thread, nice to see some positivity!
Cor, it depends on the game. I think the main fantastic thing about the genre is variety. You can expect so many things of games and get so many different results, it's wonderful to look down the list of games you have and have arcadey fun hits or deep melancholy masterpieces or loud shouty war games.
I love stories in games! Really good ones. God, you are really in a game with a good story, making the choices and fighting for the next thing to happen and pushing it along. Getting involved with characters, feeling like it's your story as opposed to one being told to you. Glorious. Something very personal about games like that. I will love Final Fantasies 7-10 til the day I die.
But I also love beautiful design and creativity. Rayman Legends is probably my game of the year with The Last of Us. So beautiful, so FLUID, really pulls out that sense of rhythm and free spirit that I get from playing a song perfectly or running around like a little kid, and all that character blaring off the screen in every wacky voice and derp face and slick art.
AND CO-OP GAMES! I've never laughed harder than in Mario Kart or trying to play through something awful like Ghost Squad, or hankering down for real intense boss battles together, or devising strategies in your living room... Real sense of unity, it's wonderful.
Games are brilliant.