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". Since the first game's characters tend to be simple archetypes, you could say Xan just is this trope. 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 II: 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 II 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.
Clive Handforth from LittleBigPlanet 2. His default position is a sad-looking slouch, and he seems eternally depressed, at least until you complete the Factory of a Better Tomorrow. From there, he's mainly monotone and snarky.
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.
The Crestfallen Warrior from Demon's Souls. He's so depressed and unmotivated that he fades away after you exhaust his dialogue options since he no longer has anything to tie him to existence.
The Crestfallen Warrior of Dark Souls is content to rest at Firelink Shrine and regale you with his pessimism and bitter "laughter" at his own downbeat "witticisms".
He eventually gets over it after you ring both Bells of Awakening, being inspired and trying to make something of himself. He promptly hits his nadir and goes Hollow in the ruins of New Londo, poor sod.
Crestfallen Saulden of Dark Souls II is just as gloomy as his predecessors, but he's notably friendlier, more helpful, and doesn't die in his storyline.
Granblue Fantasy has Narmaya: Though she's usually a fairly cheerful person, when she's sad, she's very very sad. And when someone would tell her that she's not helping, she'll mope in the corner of the room. In certain events, some characters are also aware of this side of her.
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!
In The Sims 3 and The Sims Medieval, a Sim with the trait "Morose" will be this: often in a depressed mood for no apparent reason (they literally wake up in it even if the day before went well) and harder to cheer up.
In the epilogue of Red Dead Redemption, Jack Marston becomes this after John is killed by Ross at the tail end of the game, and Abigail dies soon after.
A minor character in Undertale is a sad-sack ghost named Napstablook who likes to mope around their house listening to techno music. They attack the player with spectral tears, and can even form a hat out of them.
In MARDEK, Miserable Old Martin, one of Legion's four souls, constantly grumbles about everything.
Gozu from Shadow Warrior 2, like all of the Ancients, is an incredibly powerful immortal demon. He spends his days doting on his goddess Ameonna and moping on the front steps of the fortress doing absolutely nothing to help those below him, even as they suffer far worse than he does. Lo Wang despises him for this.
Gozu:You are always so unkind to me.
Lo Wang:And you sit up here on your self-pitying ass, waiting hand on foot for an immortal narcissist while the people you pretend to care for are victimized. Don't try to tell me you give a shit about humans. Your actions...sorry...inactions tell the story.
Machina from Dies Irae is a perpetually gloomy character who has long since crossed the Despair Event Horizon and now simply wants to die, but can't due to being bound to the Big Bad's Warrior Heaven. The narrative frequently compares his eyes to those of a dead fish and gives him the moniker of the Somber Knight.
Peatrice from The Legend of Zelda: Skyward Sword starts off like this, running the Item Check in a gloomy and complaining manner because no one ever uses its services. She breaks out of it through Link's repeated visits because she develops a crush on him, her sole customer, but she will return to her gloomy demeanor if Link rejects her advances.