Examples moved from the Sliding Scale Of Anti Heroes here, as that page is too big to acommodate them. Too many Zero Context Examples are here, so they need to explain how the character fits the trope before they can be added.
Shinji Ikari from Neon Genesis Evangelion, who saves the day several times in spite of all the mental problems.
Though in the manga adaptation and the Rebuild Of Evangelion series, Shinji's more of a Type II and sliding into Type IV at his more unstable moments.
Rock (and Benny) from Black Lagoon. The same can't be said for the other members of the Lagoon Company, though, who are Villain Protagonists.
Rock eventually subverts parts of this by being quite knowledgeable with negotiation skills make him dangerous in a different way compared to his kill-happy acquaintances. And he is eventually shown to have some serious balls despite being a Non Action Guy he becoming a Type III then in El Baile de los Muertos he has become Type IV.
Nozomu Itoshiki of Sayonara Zetsubou Sensei. AKA Mr. Despair, he is constantly attempting suicide and angsting about the most ridiculous of things. Interestingly, he isn't an example of This Loser Is You, as he's very good looking, intelligent, and comes from a very wealthy (if bizarre) family. In fact, the irony of his character is that he acts the way he does despite having these advantages.
Throughout the series', Itoshiki develops and becomes more of a Type II.
Tatsuhiro Satou of Welcome To The NHK is a highly unstable NEET who places all of the blame for his highly unstable life on a conspiracy organization known to him as the NHK.
Renton Thurston in Eureka Seven, who eventually graduated into a proper hero.
The protagonist of The Tatami Galaxy, who is something of a Zetsubou-sensei expy, and is described in some promotional materials as a "not-so-lovable loser".
Early Spider-Man, explicitly designed to be the first superhero with personal and internal conflicts besides super-villains and criminals. However his superhero alterego comes off as a Type II
Donald Duck is a good example of this trope, being This Loser Is You and not possessing any heroic traits, despite being the protagonist.
Iznogoud started out as a Villain Protagonist, but in latter comics, he tends to be arguably portrayed as this, since there are moments (especially in Iznogoud Finally Caliph and Who Killed The Caliph) where he has to accomplish heroic actions.
Yorick Brow from Y The Last Man. Whilst he is inarguably one of the most moral characters in the comic, he is essentially useless in most situations that require action, makes several bad decisions and has an (extremely understandable) case of survivor'sguilt. His heroic nature shines through when in spite of all this he still tries his best to do the right thing.
Kung Fu Panda has its main character, Po. In a strange aversion as the movie begins he is a fat, stupid clumsy goofball who doesn't know Kung fu but has good intentions. As the movie comes to a close he has become the "the chosen one one." However he is STILL very much a fat stupid clumsy goofball who marginal knows Kung fu. And as the Second movie opens he is still fat stupid clumsy and a goofball. However he is now honored as the Dragon warrior and people now respect him
Rupert Grint as Tony in Wild Target. He sinks into Type IV territory when he sticks with Victor to learn his trade even after he learns of his true profession, but both pull a Heel Face Turn when they realize that Rose is in grave danger.
He also plays Malachy (the goody-two-shoes who gets pulled by his Type IV Anti Hero friend Luke into dangerous and potentially illegal acts for the love of the lead female) in Cherrybomb.
The character of Shaun Of The Dead and his roommate Ed. Shaun subverts this by the end of film as he at leasts saves his ex-girlfriend and reconciliates with her, and he ends up surviving the whole Zombie Apocalypse.
Evelyn Waugh's first novel, Decline and Fall, has Butt Monkey protagonist Paul Pennyfeather who is one of these in the way he is rather a pushover taken advantage of by the other characters.
Rincewind is unusual in that, while he retains his constitutional allergy to danger, over time he becomes a bit of a mythic figure: It's implied that his survival instincts are nearly supernatural. At one point Death looks at the hourglass containing Rincewind's life span... only to find its shape not merely elongated and deformed, but no longer describable in human terms.
He did start off briefly as a Type V, having avaricious tendencies, and conning his employee out of his own money and betraying help from him, but then he slid gradually down the scale as the books passed before settling at Type I; his anti-hero traits softened, although he's no less cynical than he was at the start.
The narrator of Notes from the Underground is one of these, as is Franz Kafka's Josef K—in fact, in an existentialist novel, the hero is likely to either be this or a Type V (Meursault of The Stranger is a good Type V example).
Gilbert Norrell of Jonathan Strange And Mr Norrell, while a skilled magician, is a humorless and petty character who is far from evil enough to be an Evil Sorcerer, but also far from sympathetic (or interesting) enough to be a traditional hero.
John Le Carre's spymaster George Smiley is like this as a contrast to James Bond, living in the more cynical side of the Sliding Scale Of Idealism Versus Cynicism, and as opposed to Bond being stylish and a Chick Magnet, Smiley dresses poorly and is a cuckold. Smiley also invites comparison to Harry Palmer also intended as a deconstruction of Bond but who is definitely a Type II.
Lily Bart from Edith Wharton's House of Mirth. Let's see: fails at anything and everything she tries her hands at? Check. Only ever succeeds at alienating the few people who genuinely do care about her? Check. Is a whiny, insufferable Jerk Ass with an entitlement complex bigger than Brazil? Check. Dies at the end? Check.
Lola from Kit Whitfield's Benighted is pathetic, self-loathing and self destructive, turning away from or turning on anyone who might help her.
Mick "Brew" Axbrewder from Stephen R. Donaldson's Man Who series, a self-pitying alcoholic who makes Thomas Covenant look like Binky the Clown.
Linden Avery in the second Chronicles Of Thomas Covenant trilogy. Becomes a more standard heroine in the third trilogy. Stephen Donaldson is very fond of taking Type Is and transforming them.
Amir, the narrator of The Kite Runner starts out as a coward hiding from his past but grows throughout the story and is redeemed to become a 'true' hero.
Dave Lister, Cat, and Arnold J. Rimmer from Red Dwarf start out like this, although Rimmer is both a neurotic loser and a smeghead. Lister once goodnaturedly described himself as a "bum", while Rimmer would call him a lazy slob. Cat was vain, self-centered to the point of callousness, and not very smart... not surprising given that his species had evolved from a single, pregnant female housecat 3 million years ago (imagine the inbreeding), and other cats considered him a moron. However, all three became more competent in the course of the series. However, they never quite lost their essential quirks, their good qualities (such as Lister's selflessness and sense of fairness) merely became more pronounced. Or, in the case of Arnold Rimmer, who had no redeeming qualities, Rimmer had a run-in with his Knight In Shining Armor counterpart "Ace" Rimmer from an alternate reality.
Travis Touchdown, of No More Heroes, a porn-obsessed Otaku without anything resembling a social life. He's also a Type V, however, eagerly slaughtering opponents and rarely showing any remorse for his killings (with some obvious exceptions, such as with Jeane.) He gradually loses Type V elements during the sequel, however, turning into a full Type I who vows to destroy the UAA for the lives they have cost.
Lester the Unlikely from the SNES game of the same name starts out as such a wimp that even turtles scare him. He does become more heroic about halfway through the game, however.