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!
Disco Elysium made me cry after a short conversation with a random man on the street. I haven't played much of it yet, so I can't speak to the majority of the content, but what I have played of it evokes more emotion than any game I've ever played before.
It has. However, don't buy it. You can easily find it in fitgirl etc. Reason being, the devs won't see a cent as they were kicked out of the studio that published the game. There's lots of articles regarding this drama anywhere.