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 very rarely cry from games, but Ori and the Blind Forest made me shed a tear or two. Ori and the Will of the Wisps made my full-on ugly cry, and I was literally distraught for days.