I've been playing a game that I can't play without the sound currently. It's called Donkey Kong Country: Tropical Freeze.
There is something about this game that I need to hear the sound. It's not necessary for any of the challenges in the game, but the experience is really incomplete without it. It is the Music. I can't believe how amazing this soundtrack is. Even though a lot of the tunes are more ambient, they still stick with you. I also love how the music changes dynamically based on whether you are in the water or not or some more action is happening. The gameplay is really good too, and even at the really frustrating parts the mood is one of smiling and laughing than growling and throwing controller.
My only complaint is that DK makes a weirdo hee haw noise when he dies that doesn't sound as good as the original Donkey Kong Country game. I miss the Donkey and Diddy Kong monkey talk from that game.
Here are a few that have really stood out for me:
Irate Eight
(Please visit the site to view this media)
Forest Folly
(Please visit the site to view this media)
Busted Bayou
(Please visit the site to view this media)
Mangrove Cave Underwater
(Please visit the site to view this media)
Are there any games that you feel the need to always have the sound on just for the music/soundtrack? A few others that come to mind are Skyrim, Mirrors Edge, Metal Gear Solid, and Final Fantasy.