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 played Heavy Rain and somehow managed to get the ending where everyone died. That kept me off gaming in general for quite a while. Brothers was mentioned here, I can recommend it too. Papo y Yo is a allegory of your relationship with an alcoholic abusive father, and it didn’t make me cry outright but was pretty heavy.
+1 Heavy Rain. Don’t expect too much gameplay wise, the emotional story is the highlight.