Well, DmC: Devil May Cry sort of got me into aggrotech, but only Combichrist though. DMC didn't do much at all with heavy metal and heavy rock - I still do not like heavy metal, but I haven't come across "good" heavy metal since most heavy metal that comes on the radio and stuff are just screamos and other crap that don't do heavy metal justice in my opinion.
I guess Sly Cooper would be a jazz influence since it stars a sly, smooth, and clever thief. inFAMOUS 2 - I don't remember the first game's tracks that well besides the ending theme - would be that sort of tribal, visceral, instinctual drum thing that pops up every now and again. The series as a whole has this rhythm thing going on either electronically or acoustically and it's just cool and adrenaline-pumping.
Assassin's Creed in general does the same as inFAMOUS, but Black Flag really nailed it with the whole pirate-themed music and shanties. Oh, I do love me some shanties and Parting Glass is beautiful.
Muramasa: The Demon Blade would be traditional Japanese music mixed with modern sounds. In a way, it has that Samurai Champloo feel that Nujabes gave.
Metal Gear Solid 3, V, and Rising: Revengeance only gave one song from Starsailor, one from Garbage and one from Depeche Mode: Way to Fall and Wrong. I never really looked into Starsailor and Garbage, but I tried Depeche Mode and never really got into them.
Anyway, despite never playing GTA: San Andreas, it sort of got me into reggae more, but in a funny way: Drum Pan Sound by Reggie Steppa. The song is so hilarious, but so good.
In the same boat or airship: Bioshock. Never played them, but I really like the vintage, old songs from the pre-40's? Just hearing them is nice; there's this charm from old songs that you don't get from more recent or new songs.
Ratchet & Clank: All 4 One's trailer song, Cradle, introduced me to The Formidable Joy. Also, Final Fantasy XIII's localized theme, My Hands, is nice too. Yes, I like Leona Lewis and Christina Perri, Ellie Goulding, Kate Nash, Regina Spektor, Paramore, and sometimes Lady Gaga.
Here's one that's the opposite: Battlefield 3's & 4's static drop, bass drop, "dun-dun-DUH!!!" BS that's been haunting their games since BF3. It was kind of cool and tolerable at first, but it's like in every trailer and not just every BF trailer, but other games are doing this stupid "dun-dun-DUH!!!" thing. Seriously? Why can't we do something else? Why can't we have more games using jazz, blues, (good) rap, ska, country music - I'm serious, here -, classical, or even varied electronica than whatever the hell this is. It seriously sound like someone dropped a dog or cat and this "tune" came out and used to pass off as actual music. The crappiest songs and pieces in the world are better than this knock-off.
Well, if you want varied music, you can always try Animal Crossing.