This character is too ignorant to realize that he is ignorant. He has so little power and knowledge that he has nothing to compare with, and thus grossly overestimate his own power, knowledge and importance. Likely to live in a Small Secluded World or be saturated in Paranoia Fuel... or both.
A character who is characterized this way is sometimes refered to as a "King of Pointland", after an old example of this trope. Pointland in the novel Flatland is not a kingdom at all, it is just a dimensionless spot of nothingness. Its only inhabitant is "king" by default since he is totally alone. He has no width, no height, no depth (neither literally nor metaphorically), no power or knowledge, and since he has nothing to compare with himself, he believes himself to be omnipotent.
Children and animals are normally excused from this trope. They can be included in special cases when their "ignorance of ignorance" is highlighted rather than simply a part of who they are — or when they have powers way beyond their maturity, so that their lack of understanding becomes a problem.
Characters who are merely Ignorant of Their Own Ignorance are very different from characters such as a Small Name, Big Ego or a Heteronormative Crusader. While the Small Name, Big Ego is narcissistically full of himself and a Heteronormative Crusader is self-righteously narrow-minded, an Ignorant character is merely naive and doesn't know any better. While ultimately innocent, he might still be a villain — often one who is tricked by smarter villains, and thus relatively easy for the heroes to turn against their master by using their incomprehensible yet efficient powers. Compare Outside-Context Villain, who exploits ignorance of even his potential existence.
As Wikipedia can tell you, in Real Life Psychology it's known as illusory superiority.
See also Know-Nothing Know-It-All.
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Anime and Manga
Neji Hyuuga of Naruto shows signs of this. He has a very strong belief in the inevitability of fate, the futility of fighting against it, and his own position as the most powerful genin in, well, the world it seems. The fact that he's preaching to Naruto of all people about bearing a burden he never asked for or deserved just proves how little he really understand about life outside of his clan's wealthy compound.
In Bleach, before Aizen, Gin, and Tosen arrived and usurped his throne, Baraggan believed that he had already conquered everything, and was incredulous when Aizen told him there are other worlds besides Hueco Mundo.
Baraggan actually knew that the other worlds existed (after all, all Hollows and Arrancar seem to have some knowledge of the world they essentially invade for human souls) but was trying to say that actually conquering those worlds wasn't worth the trouble.
A Certain Magical Index: Several characters, mostly the ones living in Academy City, believe their conflicts and world of science is all there is, completely unaware of the conflicts and world of magic.
In Calvin and Hobbes, Calvin constantly overestimates himself. (While this is quite natural for any 6-year-old, it looks very weird on Calvin since he uses language and glimpses abstract thinking on levels far above his real-life peers.)
Played with in Marvel Comics' Secret Wars. The Beyonder was a Cosmic Being who believed it was the only thing that existed until it discovered the Marvel Universe. Subverted in that it really was omnipotent, although this was later Ret Conned to be a delusion (it was powerful, just not the most powerful cosmic being.)
Green Lantern: Larfleeze of the Orange Lantern Corps might be this. For eons he was locked away in a secluded system with his treasures, content to consume anyone who crossed him until the Controllers woke him. And while he is a formidable being, having the power of an entire corps and an entire legion of orange constructs at his command, his view of things is partially shattered when he encounters the rest of the universe after so long, circumstances force him to make deals and alliances with beings just as powerful as himself. Afterward he also realizes that some beings in the universe, like humans, are better at being greedy than he ever was.
Subverted and partially played straight in The Truman Show. Played Straight in the way that Truman grossly overestimates his popularity in his circle of friends. They actually hate him, or at least don't care about him very much, and merely suck up to him because they want to be in the spotlight. Subverted in the way that Truman actually underestimates his importance in the world. He thinks he's a normal guy at a normal job in a normal town. But the town is actually a Small Secluded World, and Truman himself is "on the air, unaware", having millions of fans without knowing it.
Also subverted in that he is Properly Paranoid and has always had a sneaking suspicion that all was not quite right, something only more evident to him as the movie progresses. He's not totally ignorant of what his friends and family think of him either- he asks his wife, at one point, why she married him, when "you can't stand me". Plus, one of the hints he gets that he's being spied on is that random strangers know who he is.
The male lead of Blast from the Past has quite a bit of this, having grown up in a Small Secluded World and not getting any real experience of the outside world. For example, his very heteronormative upbringing doesn't make him a Heteronormative Crusader — instead, he doesn't consider his gay friend to be strange at all, not having any division between heterosexuality and homosexuality included in his worldview.
It's more that he doesn't seem to realize he's homosexual, or even know what homosexuality is.
Pocahontas: "But if you walk the footsteps of a stranger, you will learn things you never knew you never knew."
In Flatland, the zero-dimensional creature known as Spot is the King of Pointland. He is the former trope namer and maybe ur-example.
Pretty much everyone but the main character applies. The one-dimensional world doesn't realize the existence of a two-dimensional world. The two-dimensional world denies the existence of a three-dimensional world, etc. The book is actually supposed to be written by the square to the three dimensional world to tell them that there are still further dimensions; in effect, an accusation of this trope aimed at the readers themselves.
The Hunger Games: Effie Trinket. Only Effie could think that she had made the perfect remark when saying "When you put enough pressure on coal, it turns to pearls!" Uh...no, Effie, it doesn't. It doesn't help that she prefaced the statement by saying she had thought up a very clever retort.
Several books by Edgar Rice Burroughs feature the protagonist visiting a Wacky Wayside Tribe where the residents firmly believe theirs is the grandest civilization in the world. Invariably, their mighty empire consists of a few pathetic mud huts.
In the post-Apocalypse novel Malevil, the evil priest Fulbert is ignorant of horses and firearms and it painfully shows. He can't get Emmanuel to give him a cow so he trades for it: two horses and three guns.
He doesn't understand the labor and military importance of horses in their new world. He gives Emmanuel two unruly mares and keeps only the tame geldings, not recognizing that Emmanuel has the only stallion in the region, now all the breeding stock as well, and his trade gives him a total monopoly on future horses.
He's clueless on firearms and tactics. On Fulbert's orders, Gazel unlocks the armory and gives Emmanuel his choice of three. After checking on ammunition, Emmanuel proceeds to grab the scoped rifles. With conflict pending between La Roque and Malevil, Malevil being a fortified castle with high walls, Emmanuel is robbing Fulbert of his best weapons and gaining a massive tactical advantage: Malevil's rifles will pick off Fulbert's shotguns long before they're a threat.
Lampshaded by his goon Armand, when he's trying to backmail Emmanuel over the horses coming with their saddles:
"Anything you can't eat, our Fulbert doesn't know the first thing about it".
Discworld: Bloody-Stupid Johnson is this ten-times through. He doesn't know he shouldn't be able to make a circle where Pi is just 3 or that it shouldn't be possible to make some of the things he makes. His works double as Achievements in Ignorance. Even the ones that don't break the laws of physics, like Ridcully's bathroom.
Chris Fogle in The Pale King, during his college years. He later realizes just how idiotic and lazy he used to be.
In The Island of Doctor Moreau, the Ape-man believes himself to be perfectly human (a "five-man") because his hands have five fingers, a trait he shares with the two genuine humans on the island.
In the Xanth novel Faun and Games, protagonist Forrest Faun gives a show of this early on when he gets a ride from Nimby the Dragon Ass and his girlfriend Chlorine. Chlorine mentions that Nimby's magic Talent is to let anyone who travels with him be whatever they want to be (which is why she is Mary Sue-grade beautiful, smart and nice). Forrest wisely conclude to himself that Chlorine is utterly insane, since Nimby has already demonstrated his singular Talent of walking up and down walls. Little does Forrest know that Nimby is actually the secret identity of the major demon XANTH, who doesn't have to play by the same rules as ordinary mortals. It's also fairly rude to make such a dismissal since the land of Xanth contains plenty of magic objects and plants that let you do exceptional things without having to use a specific Talent.
In the Honor Harrington series, most of the Admirals of the Solarian League are so convinced that they are the greatest leaders of the greatest military force in the universe that they keep walking into unfortunatesituations that they probably could have avoided with a little thought.
The Rebels in the penultimate books of Galaxy of Fear. They are quick-grown clones and have trouble telling the image of something from the reality.
In general, wizards in The Lord of the Isles series are hugely powerful, have no clue about the nature of the forces they're wielding, and are completely unaware of their ignorance. Consequently, major spells tend to have unexpected (and often disastrous) side effects: for example, in the backstory of the series, a spell cast to sink an enemy fleet beneath the seas also sank the island the spellcaster was on. Oops.
There's a case for House being this in "The Doctor's Wife" episode of Doctor Who.
Dean Pelton in the Community episode "Conspiracy Theories and Interior Design". Jeff quickly realizes that the Dean isn't capable of forming a conspiracy on his own and actually fails to understand what a conspiracy is with the Dean in the end 'conspiring' with everyone involved.
Jethro from The Beverly Hillbillies. Assumed his sixth grade education made him a genius and what didn't explicitly "know", he could figure out. The rest of the Clampetts, lacking even that much formal education, consider him merely a Know-Nothing Know-It-All. Which is still leaps and bounds above the buffoon he really is.
In moral philosophy, one classic thought-experiment is the one about Happy Bert. This guy is just Sarcasm-Blind, incorrectly believing that the people laughing at him are actually laughing with him.
Religion and Mythology
Several of the most famous deniers of Evolution fall into this trope. Most notably, Australian businessman-turned-fundamentalist preacher Ken Ham, who owns and runs the Creation Museum, which teaches how early humans cavorted around in friendship with the dinosaurs. His lectures are filled to the brim with misinterpretation of basic biology, up to and including misinterpreting Evolutionary Theory, plus misunderstandings of physics, chemistry, astronomy, geology, cosmology, and psychology. But please don't bother telling him he's getting it wrong, because he'll only start lecturing you on why he's right and you, in fact, are wrong.
St. Augustine warned against this type of activity, when he was debating creationists in the 5th century (making the trope Older Than Feudalism). Specifically, he warned: "Usually, even a non-Christian knows something about the earth, the heavens, and the other elements of the world, about the motion and orbit of the stars and even their size and relative positions, about the predictable eclipses of the sun and moon, the cycles of the years and the seasons, about the kinds of animals, shrubs, stones, and so forth, and this knowledge he holds to as being certain from reason and experience. Now, it is a disgraceful and dangerous thing for an infidel to hear a Christian, presumably giving the meaning of Holy Scripture, talking nonsense on these topics; and we should take all means to prevent such an embarrassing situation, in which people show up vast ignorance in a Christian and laugh it to scorn."
GURPS Aliens include a hivemind race called Mmm. When first encountered, this creature slaughtered humans just to see what they looked like on the inside. However, Mmm just didn't know any better: Never before having encountered a race of individuals, Mmm assumed that humanity was a hivemind and that it wouldn't mind losing a few drones to Mmm's curiosity. (When it found out the truth, it was appropriately horrified.)
The Ender’s Game universe contains basically the same scenario, but worse. The aliens were assumed to be Always Chaotic Evil since they slaughtered all of the humans left alive when they captured their ships; it later turned out that they were a Hive Mind, and assumed humanity was the same way. They thought they were just cutting off communications.
In the Touhou fandom, Cirno tends to get this treatment. While she was portrayed as somewhat stupid in her first appearance, it has been wholly embraced in subsequent appearances, declaring herself "The strongest!" after bumbling through a few fights in one, and trying to take on what appeared to be a Giant Mecha in another.
To be fair to Cirno she says she's the strongest after winning a match and as a playable character she IS actually capable of defeating EX level bosses. According to Word of God however she is still an idiot.
Cirno only claims to be the strongest fairy in Gensokyo. While this is technically true, most fairies are Cannon Fodder that the player character defeats by the truckload, so this isn't saying much.
Zinnia Jones: In the episode "Filling in each other's blanks: The importance of listening", Zinnia discuss the concepts of "known unknowns" versus "unknown unknowns". With "known unknowns", we are least know what kind of answer we are looking for. That something is an unknown unknown, however, means that we are even ignorant of our own ignorance. Zinnia argues that we should always listen to people's experiences: If nothing else, it can still alert us to questions we didn't yet know we needed an answer for.
Metamor Keep: In the story "Keeping The Lamp Lit" Lord Altera Loriod becomes the living embodiment of this as he's seemingly unaware that males who used to be females would not have any knowledge of how combat actually works in the middle ages, believes that Goldfish are made of real Gold no matter how many times he has been told otherwise, ignores everything the spirits tell him, and believes that if you rape someone in a cursed area that you have kidnapped they will get horny and turn into a woman instead of feel helpless and turn into a child, when he is. and in a later story by a different author "Where Life Begins Anew" the Shapeshifter known as Hawl Enroygall can't seem to grasp that it doesn't matter if you can disguise yourself as anyone on the planet it won't work if the only person you can act like is yourself. Even Cedric Bariclauph sees through this and he had not even met Hawl.
Cracked's Gladstone lists 4 Douches Who Amazingly Don't Seem to Know They Suck. These include people who use a handicap tag for parking without actually needing one, people who grow weary of their customer service job, drivers who honk at other drivers merging onto a highway, and people who overreact to minor inconveniences.
On SpongeBob SquarePants, Patrick Star manages to embody this trope while simultaneously applying it to someone else.
(Spongebob pretends to be stupid. Patrick immediately forgets that he's pretending.)
Patrick: Dumb people are always blissfully unaware of how dumb they really are. *stares blankly and drools a little*
Known amongst psychologists as part of the Dunning–Kruger effect. Your ability to judge how good someone is at a skill is directly proportional to your ability at the skill itself, barring outside instruction. In the dearth of other information, most people will assume that they are above-average (around the 66%th percentile) in ability, regardless of their actual ability level, with very competent people believing that most people are more knowledgeable than they truly are, and the incompetent people believing themselves to be far more competent at the task than they actually are. The only way to actually judge someone's skill at something is to have someone skilled at the task do so. Naturally, incompetent people don't believe that they are incompetent, and think that they can judge the competence of others...
This trope is also the reason why so many cases of Insane Troll Logic or outright bigotry are accompanied by the words "Sorry, but it's the truth..." or "I'm just telling it how it is!". Although in some cases, the poster may simply be trying to convince him/herself of their own wisdom.
There's a Spanish proverb that says: "Tell me what you brag about and I'll tell you what you lack."