I can't really think of any original songs that haven't been mentioned already (you're killin' it, GAF!), but I do have to give Telltale props for their godly choices of licensed music for both their openings and closing credits. Whilst Tales from the Borderlands has the incredible opening song choices, The Walking Dead games that tend to have the poignant closing credit music. Special mention goes to TWD S1's Take Us Back, S2's In The Pines, Michonne's catchy Wolf and the chilling To The Bone. Tales from the Borderlands also had Silver Lining at end of the series. Thanks for getting me addicted to First Aid Kit, Telltale!
A lot of good responses in here, but my personal favourite is the Max Payne 2 credits. This song (Late Goodbye by Poets of the Fall) was written specifically for the game: