Basically what the title says. I've only played two games like this so far (Silent Hill 2 and OMORI) and am looking for more.
Could be games, ROM hacks, mods of existing games, etc.
EDIT: So many great suggestions! Getting a bit overwhelmed by how many there are, so I'm going to need to find a way to put this in some kind of list with an order to play the games in. By all means, keep 'em coming, though!
I may be a bit late to the party but The last of us 1&2 had a pretty emotional impact on me. The game is pretty brutal, in every meaning, but the overall experience and story telling is hard to top.
I highly recommend to not get spoiled and play it in order of release!