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Xan, the morbidly depressed elven enchanter from the first Baldur's Gate game. A typical line is "Our Quest is in Vaaaain" and "Oh what is the point". He is later referenced in the first Icewind Dale game when finding a diary of an elf, who feared he "Had become as miserable as my cousin Xan".
Nearly every non-player elf character in the add-on Dungeon Siege 2: Broken World is morbidly depressed, especially Amren, who whines about the loss of his clairvoyant powers every time you talk to him, no matter what topic you wanted to discuss. Considering what crappy state the world is in due to the cataclysm after the end of the second game, they may have a point, but it quickly gets annoying.
The dryads... well, most of the dryads have just gone nuts, but they have to cope with the loss of their forest and most of their population, so their destructive mood is at least justified.
In Star Control 2 the Utwig, after losing the Ultron, their source of motivation, are an entire species of Eeyores, to the point that they ponder whether to commit mass suicide with a giant bomb, but decide not to because they deserve to suffer. They get better once you fix the thing. In the mythical third game, they go Eeyore again when they lose the Ultron; one of the few good ideas the game has is that when you return it to them, they become audibly happier.
...Even though they express emotion through Masks.
Ugh. I suppose, as a courtesy, I should extend an appropriate greeting. On behalf of the Utwig Proctors I truly hope, for your sake, that your day has been better than ours, although this really isn't saying that much.
In Wing Commander II, Lieutenant Etienne "Doomsday" Montclair. Almost every comm message he ever sends you is about his impending death. ("I just bagged a Cat. I bet I'm going to die now.") Ironically, as of Wing Commander Arena he's still alive, and still actively flying as per the manual, Star*Soldier.
Also played straight in the Wing CommanderExpanded Universe. Notably, he survives the entire war, including several campaigns that had single digit survival rates for everybody involved.
Despite the fact that he's the main villain in the Mario Series, Bowser sometimes makes a good example of this trope.
Negative Man from Mother 3. He rarely ever attacks, and if he does, it'll do Scratch Damage. Most of the time, he just goes on about how pathetic he is, and that there's no way he can win.
Schwer-Muta Casasola Merkle note the German word "Schwermut" means "gloom" of RosenkreuzStilette fits this trope. Basically, she always carries a depressed look on her face, she's reclusive and distrusts everyone she meets, and not to mention she's also very shy. She thinks her only friend is her pet squid, Zeppy.
Elk from the .hack// series has a gloomy face as his default expression.
Elcor from Mass Effect seem like this, always speaking in a flat monotone... but it's a subversion, because humans are simply unable to physically recognize their emotions (elcor express emotions through pheromones and bodily movements other species are unable to sense). That's why they start every sentence to non-elcor with their current emotional state. The all-elcor version of Hamlet went over pretty well with humans in-universe though.
Fhjull Forked-Tongue from Planescape: Torment is a baatezu, a being Made of Evil, who's forced (because he's also made from pure order) to keep a Deal with the Devil that forces him to only perform saintly and good deeds. Note that at no point does the contract force him to act saintly and good, leading to a perpetually angry, annoyed and surly devil who does whatever's asked of him through gritted teeth.
The Nameless One: I need a place to sleep. Fhjull: WHAT? I must endure your presence even longer? Will this torment never cease? Feh! Fine, then, fine! Spread out across the floor! Raid my larder! Knock over my experiments! I'll just be over here fuming! Feh!
Zor from Sonic Lost World looks and speaks like a depressed goth/emo. He even seems to hate a lot of things.
Sigh... bad news is actually the one thing I enjoy...
Sullen, the magic mirror in Magic Fairy Tales: Barbie as Rapunzel...although he is the unwilling pawn of an evil sorceress, so it may be justified.
Barbie: Isn't it a beautiful morning, Sullen?
Sullen:(sighs) If you like sunshine, I suppose. The glare gives me a headache.