This is list is incomplete, entirely subjective, and presented in no particular order. Most of the music here is from games I personally played last year. I'm sure there are other excellent soundtracks from 2013 and early 2014, so let me know if you have any other recommendations!
A lot of the credit for this list goes to the Top Score podcast by Emily Reese, over at Classical MPR. Additional inspiration from Kotaku.
God Only Knows
Will the Circle Be Unbroken