Ironically, the one time he's in a relatively good mood and tries to cheer Asuka up, this ends up being exactly the last thing Asuka needs, as her confidence and sense of superiority has been broken and having the unconfident Shinji trying to cheer her up pushes her to the brink.
Then there's Rei Ayanami, whose very existence is an insult against the Laws of Nature and all that is right and good in the world. She can be accurately described as Gloominess Personified.
Sugata from Star Driver definitely fits this, going as far as inadvertently hurting Wako with with his downer behavior. However, considering that he can't leave Southern Cross Island due to being the bearer of Samekh (if he does, his Ninja Maids will be forced to kill him), is also the bearer of a power that can hurt or kill others and put him into a coma, and has to live in fear of Wako's seal being broken and/or her being harmed, this is pretty much justified.
Played for Laughs in Sayonara, Zetsubou-Sensei with Nozomu Itoshiki. It doesn't help that his name, written in a certain way, literally translates as "zetsubou-sensei", or "Mr. Despair".
Ryoga Hibiki of Ranma ½, who legitimately has a lot to be sad about, is nevertheless prone to such heavy depression he's able to use it to fuel a ki attack - one so powerful it leaves a massive, smoking crater and frequently demolishes his opponents in one go. Yes, that's right, a literal Angst Nuke in the form of a Kame Hame Hadoken.
Miranda Lotto of D.Gray-Man, thanks to years of failing at every single job she attempted (due to being a Cute Clumsy Girl taken to absurd degrees) slowly eroding her sense of self-worth, combined with spending a long time stuck in a "Groundhog Day" Loop. She gets better when she finally finds something she's good at in becoming an Exorcist, but still lapses every now and then when she thinks she screwed up.
Cardcaptor Sakura: Even after revealing their soft side, Syaoran and Yue are still rather humorless and snarky in tone. Spinel Sun also seems to count as one.
When Crona shows off his/her poetry to the other cast members, it turns out to be so depressing that it makes everyone who reads it sit in the Corner of Woe and wish they were never born.
Kaoruko of Comic Girls tends to think things very negatively. For example, the fact that she messes up a few pages of Tsubasa's manuscript in the first episode already makes her consider killing herself.
Ishigami from Kaguya-sama: Love Is War was rarely happy early on in the series, and it even made a Running Gag of him wanting to go home and die. Later chapters would delve into why he had that mindset, as well as help him overcome it.
Yuuutsu-kun to Succubus-san: Yuu starts the series in a state of serious depression because of how badly he's been overworked. When Sakuma makes him leave that job, he spends most days literally lying around doing nothing, or occasionally sitting and watching the outside through his window. Sakuma's entire goal is to help him get better, and there are consistent signs that he values her presence in his life. Yet, even as he improves, Sakuma can't imagine what Yuu smiling would look like. When she tries it comes out as a blurred image.
Izumi Suni from She doesn't know why she lives. is an absolute mess of anxiety-induced pessimisim, and she tends to assume the worst any time she makes a minor slip up or misinterprets an innocent statement.
Zonze Ge from Magu Chan God Of Destruction is not only perpetually depressed, his power actually causes extreme depression in those he comes in contact with, a fact that makes him even more depressed.
Shota Aizawa from My Hero Academia. He’s often pessimistic and negative with his students, though it’s actually because he cares deep down, and tends to be apathetic enough to let a couple of students teach while he sleeps in his sleeping bag in class when he doesn’t consider a topic important. He lost a close friend during his own time as a student and likely has some degree of clinical depression.
