<span class="bold">Immortal Defense</span>: tower defense. You're a human god of deep space created to fight off alien invaders. Fight other gods, try to not go nuts, fail. Better than Planescape: Torment. Unfortunately, the new and improved version is Steam only, but it's one game worth getting Steam for.
Anodyne: top-down action/exploration game with JRPG-style one-sided "dialogue"; a feeling of inescapable doom guaranteed.
<span class="bold">Spirits</span> (70% off on Humble now, DRM-free): Lemmings but fluffy, slower-paced and saaaaaaaad.
Schein: clever puzzle-platformer. You're a guy looking for his lost child in the swamp, helped by a friendly and ever so slightly sinister ghost lady. Excellent writing, thematically appropriate mechanics.
The Last Door 1+2: point-and-click horror game. Everyone in it is messed up.
The Shivah: point-and-click adventure. You're a Rabbi, and a former member of your congregation may or may not have gotten murdered.
Rogue Legacy: roguelite platformer with permanent powerups. Everyone in your lineage is cursed to die trying to complete a pointless quest.
Anvil of Dawn: first-person exploration "RPG". Travel through the land ravaged by an evil army and try to assassinate the immortal invincible warlord.
<span class="bold">Glum Buster</span>: It's a... actually, see for yourself, it's free (well, donationware). I cosplayed as the protagonist once.