Adrian Mole believes he is a gifted author and celebrity chef. In reality his unpublished work is terrible, his only published book was written in his name by his mother, and he was once the presenter of a low-budget cable show about how to cook offal. All of this goes right over his head as he tries to use his "celebrity" status to his advantage; and frequently writes to people who are actually famous to ask for favours (such as to speak for free at the Christmas dinner for his book club), ask for his own show on radio, or to offer insulting suggestions about their lifestyles.
Dead Famous by Ben Elton has a few examples of the trope. Layla believes she is smarter and more attractive than the other House Arrest contestants; but is the first to be voted off the show because the others hated her. She also describes herself as a fashion designer but is just a shop assistant. David proclaims himself to be an incredibly gifted actor, to the point of hypocritically berating Kelly for "prostituting" her talent by appearing as a movie extra; but in reality his career is going nowhere and he lives off his earnings as a porn star. Geraldine thinks that Kelly herself is this, and thus resolves to destroy Kelly's public image through Manipulative Editing.
As Geraldine points out in her "The Reason You Suck" Speech after the murder, all the House Arrest contestants are this to varying degrees; they're all convinced they're witty, glamorous and attractive people who all deserve to be famous, whereas Geraldine quite rightly informs them that they're actually just a bunch of self-obsessed, bitchy narcissists who have somehow deluded themselves into believing that they're entitled to fame despite having very little talent and being no more interesting than the thousands of other cookie-cutter reality show contestants who've came before them.
Diary of a Madman's Poprishchin, the eponymous madman, believes himself to be an important person, that his holding an unimportant position at age 40 is non-indicative of his career, and those trying to dissuade him from pursuing the director's daughter are just envious.
The main character Greg imagines that he'll be famous one day, even though he has no skills to make him popular.
Krisstina the singer calls herself internationally famous, but Greg doubts that she's even performed in another state.
Discworld's Lord Rust believes himself to be a fine military commander despite ignoring such factors as provisions, location, and the respective size and experience of forces. When justifying his tactical decisions, he frequently cites battles when the people in that scenario lost, or on a couple of occasions, were nursery stories.
General Ashal: I believe the motto of his old school was "It matters not that you won or lost, but that you took part."
Prince Cadram: And, knowing this, his men still follow him?
He shows up several times in the books, usually to be a thorn in Vimes's side. He, like many nobles in the past, believes himself to be superior in every way to the common rabble when really he's an Upper-Class Twit.
Lorenzo Smythe, the main character of the Robert A. Heinlein novel Double Star. While he is a very good actor (enough that the Emperor says that he's Smythe's biggest fan), he considers himself to be an artist par excellence, and the standard by which all other actors should be judged. He prefers to refer to himself as "The Great Lorenzo." When he goes missing, not even his agent notices for a long time. His obituary only mentions that he hadn't had a job in months, and that he was likely Driven to Suicide by depression.
Several early villains in The Dresden Files, most notably Victor Sells, Leonid Kravos and the unnamed "Darth Wannabe" in Day Off. All of these are people who think that that lucking into a smidgen of magical power gives them the power to take on a guy who... let's just say that at least one species of supernatural predators has ended up extinct because they managed to get themselves onto his shitlist.
This is recurring theme in Flatland, as it was intended as a satire of Victorian attitudes, particularly the Cultural Posturing of The British Empire, every "Land" believes itself to comprise the entire universe, utterly ignorant of higher dimensions. Pointland however, takes the cake, it is a single sentient speck floating alone in a void of nothingness—the speck comprises Pointland itself, and its sole inhabitant, king and self-proclaimed God. All of those are the same entity by the way, simply existing in nothingness while constantly singing its own praises. It is visited by a square and a sphere, but when the square tries to talk to it, the Point assumes that what it is hearing is merely an aspect of itself, and it begins to wax lyrical about how awesome it is to be able to have a conversation with itself, because it simply cannot conceive of the existence of something other than itself.
Gilderoy Lockhart in Harry Potter and the Chamber of Secretsis a big name, but takes credit for what others have done. Of course, it could just be an act designed to keep his fraudulent reputation afloat. His vanity about his physical appearance is almost certainly genuine, though. J. K. Rowling has admitted that Lockhart is one of the few characters in the series explicitly based on a real person. While refusing for obvious reasons to reveal that person's identity, Rowling has said he was actually even worse than his fictional counterpart and suggested that he's out there now claiming to be the inspiration for Dumbledore, or that he wrote the books himself and just let her take the credit out of the goodness of his heart. As his actor, Kenneth Branagh, put it, Lockhart "feels himself to be terrifically important, thinks of himself also as being terrifically modest. He is neither of those things."
Owen Thyll in A Harvest Of War, the Big Bad's son and heir as well as plant as "mayor" of Draeze to aid in subduing the city. He's so useless his own mother casually discards his life.
Zaphod Beeblebrox from The Hitchhiker's Guide to the Galaxy, who's actually a pretty big name (he became President of the Galaxy because he wanted to steal a ship, created the most potent alcoholic drink in the universe(s), then saved the universe a few times. Mostly while drunk) with a truly colossal ego:
If there's anything more important than my ego on board this ship, I want it caught and shot!
The Restaurant at the End of the Universe introduces a device called the Total Perspective Vortex, a simulation that drives people insane by starting on an image of them, then pulling an Epic Tracking Shot that shows them how tiny and insignificant they are when compared to the totality of the universe. Zaphod becomes the only person to ever walk away unaffected, because as he saw it, it showed him as the center of the universe, which only validated what he already believes. Oddly, his "most important person in the universe" status is a Justified Trope: This was in a fake universe which was created specifically to shelter him, so in there he was the most important person in the universe.
The Magic School Bus in the Ocean features a character named Lenny the Lifeguard, a good-natured but somewhat arrogant lifeguard at the beach, who is first seen showing other beach goers pictures from his 'daring rescues' (which, judging from the pics, weren't that daring). He ends up The Drag-Along when he sees Ms. Frizzle drive her bus into the ocean, getting swept up in the latest field trip. Throughout the story, he tries to maintain an air of authority, despite being pretty much redundant. Once the madness ends, he is elated to have "saved a whole class."
Lady Catherine, for example, is not always treated with the automatic reverence which she seems to expect (except by Mr. Collins).
Speaking of Mr. Collins, the reason he isn't able to accept Elizabeth saying "no" to his proposal and insisting that she must be playing hard to get, is that he believes he's such a fabulous catch for her that he cannot conceive of her not wanting to marry him.
Mary expects that everyone will want to hear her sing and know her opinion about every subject, but she's treated as just as silly as her younger sisters.
Darcy at first appears to be a Subversion of the trope (since he has close friends even if his casual acquaintances think he's full of it), but we later found out he's not as conceited as he first appears to be.
Mrs. Bennet's fretting about what will happen if her daughters fail to find suitable husbands before Mr Bennet's death tends to focus more on what the effects will be for her more than anyone else.
Lydia might have this the worst, however, since she treats her elopement with Mr Wickham as a big game for her own benefit despite the fact that it could lead not only to her ruin but the ruin of all her sisters, and displays nothing but simpering glee at her 'good fortune' once she's actually married.
Cersei Lannister is a rare female example. She believes herself to be, among other things, clever, ruthless, and irresistible to men. Of those qualities, only 1.5 are true. Unfortunately, she's the queen of Westeros, leading to some problems in her reign.
Mace Tyrell is Lord Paramount of the Reach and Warden of the South, so he is a pretty big name, and has an ego and pomposity to match. However, he's mostly a puppet for his mother, he's widely regarded as a joke, his one major accomplishment is due to the work of his vastly more competent bannerman Randyll Tarly, and he receives only the bare minimum of respect required by etiquette from everyone, including his own children.
In The Tower Of Geburah by John White, we have Theophilus, a flying horse with an ego the size of a small mountain. Among his claims are that Gaal (the Jesus-allegory of that fictional universe) often consults him for advice. Theophilus's arrogance nearly gets the female primary and secondary leads killed when he deviates from the flight path Gaal told him follow, causing the three to be out after sunset (until the Big Bad is defeated, flying after sundown is freaking dangerous).
All the other Clans seem to believe Firestar is this due to his tendencies to try and talk things out instead of fighting and the fact that he's helped each Clan in the past; the other leaders consider him arrogant.
In Bluestar's Prophecy, Thistleclaw and Oakheart are both said to be arrogant pricks...though Oakheart later turns out to be a Jerk with a Heart of Gold.
High Lord Weiramon from Robert Jordan's The Wheel of Time novels. He thinks he's charming, witty, the perfect general, and a clever enough liar to make the protagonist think he's totally loyal to the cause. He's actually so unbelievably suicidally stupid that some fans think he's actually The Mole, sabotaging the goodies from within by "accidentally" screwing everything up.
A character who was never seen alive, Captain Darillian, was "a petty guy who reached his ultimate level of usefulness driving a minelaying barge for a warlord and then had to be scraped off the floor." His ego was big enough that he kept a Captain's Log in full holo and talked into it like he was always on dangerous missions upon which the fate of his sect of the Empire rode.
His boss, Admiral Apwar Trigit, was also an example, albeit a much quieter one. He fell for every single trick the Wraiths set up. Aaron Allstonaddressed why in his FAQ.
Because he's not as bright as he thinks he is. He's creative in certain intelligence-gathering functions, but that has led him to believe that he is brilliant at everything. It's this assumption of his own infallibility that leads him into several errors.
And then we have Trigit's boss, Warlord Zsinj - although with Zsinj, it's a form of Obfuscating Stupidity which he knows a lot of people can see through. He's certainly got an ego, but it's not as unrestrained as it seems.
In the second novel, the "commanding officer" of a crew of repair technicians on a fully automated power relay station takes his job way too seriously. When the Alliance invasion fleet appears in orbit and commandos hijack the power relay network, he tries to rally his crew to join the defense of their homeworld, until they explain to him that there is nothing they can do to shut down the station, that it doesn't have any kind of weapons, and he should just relax and join their card game.
Peter Pan is very cocky. Wendy notes of this very early on.
Peter: (After Wendy has sewn his shadow back on) How clever I am! Oh, the cleverness of me!
Kristy from The Baby-Sitters Club has a tendency to overestimate how important she is to the eponymous club, and how important she and the club are to people in general. In one book, she's babysitting for her siblings when a friend of her brother's she's never met before comes over, and the friend's parents are hesitant about leaving their son with a thirteen-year-old they don't know. Kristy mentally fumes about this, don't these people know who she is, she is president of The Babysitters Club!