So at this point, I don't think there are any more 360 games I'm likely to get. I should mention I'm counting things I bought through XBox Live Arcade as separate from this list. That's for games I can't get in physical form for the console. If I really want to play it, and there's a physical copy, I'd buy that.
Given those criteria, I've owned 38 games for the system, though I only have 24 of them these days. We can safely eliminate the other 14 from consideration, so what did make the top 5?
1. The Saboteur
3. Tales of Vesperia
5. Alice: Madness Returns
So, GTA clone, first-person stealthish game, JRPG, first-person shooter, action-adventure/platformer game. Pretty much all of them take place in worlds that are either already ruined, or one their way there. Alice's mind is falling apart. Certainly the altered timeline Ranko creates is a mess, and even before that, Katorga is a horror show. Vesperia's world is one where humanity's ability to survive is contingent on their ability to maintain any barrier they can create between their cities and the outside world, and their method will destroy them anyway. Dunwall's on the verge of collapse, and Sean's running around in a Nazi-occupied country, which I think speaks for itself. I know most games take place in worlds with problems, otherwise what is the character doing, but this selection seems a bit grimmer than most of the previous consoles. Probably says something about my state of mind I'd rather not consider.
I haven't tried to go back and play The Saboteur through again to see if it holds up. I kind of suspect it won't. I've never played all the way through any of my other GTA-style games more than once, either. But this one just had a style that worked for me, and it made me think of Duck, You Sucker in certain ways, at a time when that movie was on my mind a lot.
Until I got to Tales of Vesperia, there was a long stretch where I had no idea what #5 would be. It was kind of going back and forth between Catherine and Enslaved: Odyssey to the West. But the former struggled due to my issues with puzzle games, and with how much I despised Vincent, and the latter had some plot developments I found distasteful, plus I wanted more freedom to use Monkey's Cloud. The console was kind of shaping up as a system with few games I really loved, mostly just a lot of OK games, each with a few different strong points, but also some weaknesses. Since Tales of Vesperia, I'd say both of those games have been passed by Sleeping Dogs, South Park: Stick of Truth, and probably one of the Metro games, though I'm not sure which (leaning toward Last Light because of the gameplay).