Borderlands 2 and Tales from the Borderlands were really great stories. Tales is a classic Telltale game, so really any of those should be on a story driven list, but Borderlands 2 deserves far more credit than it gets. Sure, they don't take themselves too seriously, and have no problem relying on sophomoric fart jokes, but the character development and Plot twists are fantastic. There were points that I honestly started to tear up. (Sorry, Bloodwing) and 2 minutes later I'm laughing my ass off because Jack can't find his violin. (He had this whole thing planned out, it would have been great.)
Sorry, I got deeper into that than I expected.
Of the less mainstream, What Remains of Edith Finch. The less I say about this one, the better. Very dark, very morbid. Only takes a few hours, but it tells some disturbing stories.
Also Inscryption. I thought I was just getting into a card game, but there is SOOO much more to the game than that. Trust me, if you do play this game DO NOT GIVE UP ON IT until the WTF moment happens. If you don't know what that means, keep going, you haven't seen anything yet.