A character always appears wearing a headdress of some kind, and never, ever takes it off. It doesn't have to be a hat, any kind of headgear is fine. Furthermore, it doesn't always have to be the same headgear, the point is that this character is never seen bare-headed. This gives him a certain mystique, since while it might just be that he wears the hat because he feels like it, the fact remains that we never learn what his hatless head looks like.
Please note that this is not about characters who usually wear a hat. It's for characters who always and without exceptions wear one, even though they don't need to. Maybe exceptions can be made if the character is seen hatless just once, but no more than that, or if it happens very rarely and is made a big deal of. A good rule of thumb is that if it doesn't seem weird, when you think about it, that the character in question always wears headgear, then s/he doesn't qualify. So obviously a character that appears briefly in two episodes of a threehundred-episode series doesn't qualify, even if his fedora was on all the time.
In some cultures, a hat is a mandatory article of clothing — something you would no more leave off than you would leave off wearing pants. This may be required for only one sex, or for both.
Please Keep Your Hat On is related. Mostly overlaps with Nice Hat. Sometimes overlaps with The Faceless.
Examples:
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Anime and Manga
In Bleach, Urahara Kisuke wears his green-and-white-striped hat constantly. The brim usually keeps his eyes in shadow, and Urahara frequently uses it to help obscure the rest of his face. When someone knocks his hat off, Urahara is quick to put it back on, even if it has been damaged.
Jigen, from the Lupin series, is never seen without his fedora.
In Black Cat, the protagonist's partner, Sven Vollfied, never stops doing three things; wearing a hat, flirting and smoking.
From Keroro Gunsou, all Keronians have hats, usually a long-eared one with their personal symbol on it. One episode involved a notorious Unreveal when Keroro switches to a baseball cap after his hat is picked up by a toddler.
Surprisingly averted with the main character, Luffy, the only character (aside from Chopper) whose hat is actually important. His straw hat is his symbol, going so far that when he achieves fame, the World Government calls him "Straw Hat" Luffy, and his crew the "Straw Hat Pirates". It's Luffy's most precious treasure, given to him by his idol. And, as it turns out, it was once the hat of Gold Roger, the pirate king. And yet, he's constantly seen without it, typically taking it off at the start of major fights and either setting it on the ground or giving it to a crewmate.
The title character of Shinryaku! Ika Musume (seen above), who would die if she removed her hat. It's hinted that her "hat" is actually her scalp (which explains why removing it is lethal); in one issue where she was suffering from amnesia, she tried to remove it only to find it fused to her head.
Ash in Pokémon always has his trademark baseball cap on. The number of times he's seen without it is directly proportional to the number of formal outfits he's been put into over the course of the show.
Comic Books
Jughead Jones from Archie Comics is seldom seen without his signature crown-shaped hat.
The titular character of obscure Swedish comic book Bobo always wore a Tyrolean hat.
In Swedish comic Herman Hedning, Herman always wears a helmet.
Judge Dredd almost never takes his helmet off. When he does, his head is always obscured by something else. The one exception to this was one time when he'd had his face changed into a completely different one.
Every single smurf wears a white hat, except for Papa Smurf, who has a red one.
The titular main character from the British comic Buster was never seen hatless except for the final issue. Apparently, he looks just like Dennis the Menace (UK) under it.
Some runs of The Mighty Thor have depicted Loki this way, though as with many comic book characters it's Depending on the Artist (it's fairly established that his hair is short—shorter than in the film version, black, and pomaded back). The J. Michael Straczynski/Kieron Gillen era took this trope to heart, however, by depicting a Loki who wore his horned helmet and cowl even when shirtless and hammering at a sword a la The Blacksmith.
Film
The movie adaption of Carrie has the character Norma perpetually wearing a red hat. When she gets her hair done, the device she's using wears it instead.
In Fat Albert Movie, Dumb Donald explains that he always wears his hat because the animators didn't give him a face. He is forced to remove his hat and discovers that he indeed has a face. When Albert's friends return from the real world, Donald had to put his hat back on because, in their fictional world, he indeed has no face.
In Star Wars, Darth Vader always wears his helmet, since it's part of his life support system. Seen only twice without it: the first time in a special pressure chamber, the second time dying. Though he is able go without it near the end in The Force Unleashed.
Literature
In Elis List, Eli always wears a Botafoga snowcap while she's awake and a nightcap when she's sleeping.
In The Vampire Files, gangster Whitey Kroun got his nickname because he always wore a white hat, even as a teenager.
Referenced in a poem by Spike Milligan:
American detectives Never remove their hats While investigating murders In other people's flats.
PS Chinese 'tecs Are much more dreaded And they always appear Bare-headed!
Nisu Uuno from Wremja always wears a ski cap and refuses to take it off.
Unless he has a very good practical reason not to wear it, you're not very likely to see Jamie Hyneman without his beret. To the point that when a particularly fierce wind actually does blow it off, Adam is left slightly stupefied because something like that had never happened before.
Over the course of three seasons of Robin Hood, you could count the number of times that Much takes off his hat on one hand. It's actually a Tear Jerker when Robin takes off his hat for him in the very last episode. (It Makes Sense in Context).
Tenkaichi from The Conditions Of Great Detectives never takes his hat off, even when he's in the bath. One character attempts to take it off him when he's asleep but he wakes up and stops them the moment his hat is touched.
Beetle Bailey always wears either a hat or a helmet.
From the same starting artist, Chip in Hi and Lois.
For many years in Doonesbury, B.D. was never seen without his football helmet. When he was called up from the Army reserves for the Gulf War in 1991 he wore a military helmet instead. For a while he was a cop, wearing a motorcycle-police helmet. For decades he wore a helmet until it finally came off on April 21, 2004, when B.D. lost a leg while serving in Iraq. His only reflections on losing the helmet was on July 31, 2004, when he thought the page quote to himself, and when he later reveals during a therapy session that his mother was very neurotic about safety, and made him wear protective helmets throughout his childhood, which resulted in him feeling naked without one as an adult.
Swedish comic strip Elvis used to feature a character called Totte, who always wore a beret.
Harry Dinkle of Funky Winkerbean never took off his band cap, except when wearing a helmet when helping with the school's construction. It wasn't until he retired from being the band conductor that he took it off... and his face was seen.
Theatre
Fiddler on the Roof: "For instance - we always keep our heads covered, and always wear little prayer shawls. This shows our constant devotion to God. You may ask, "How did this tradition get started?" I'll tell you! ... I don't know. But it is a tradition!"
Video Games
Captain Falcon is only depicted without his face-obscuring helmet twice, if you win the GP in Master Class in F-Zero X and at the end of F-Zero GX's story mode. Even then, in the case of the latter he faces away from the camera in the so you only see the back of his head. This is why it was a huge deal when, during his Heroic Sacrifice in the anime, his helmet flies off and his face is revealed. To be fair, other characters throughout GP Legend's run were able to deduce his Secret Identity and Falcon did outright reveal himself to one character, but his back was to the viewers then.
Five of the nine classes in Team Fortress 2 wear hats by default. Four of them can be made hatless by obtaining hat-free items equippable in the headwear slot. However the Soldier can only have his hat replaced by other hats while the Spy and Pyro wear hats over their balaclavas and gas masks instead of replacing them—their models don't even have faces under them.
It would be a lot shorter to list the Touhou characters to which this doesn't apply, although in many cases the hair accessory is just a ribbon or a headband. Fanon has taken this concept and run with it—a yukkuri without an accessory is at best shunned by other yukkuris but is more commonly outright attacked by them.
Brain Dead 13: Averted in Vivi's Salon, when Vivi grabs Lance's baseball cap and uses it as a gag to ensure that he can't scream during the painful "makeover".
Also averted in some death scenes.
Webcomics
Black Mage in 8-Bit Theater actually does remove his hat once, though it's not shown on screen and the only person who sees it is driven mad from the sight.
Jameson from Girls with Slingshots is always wearing a bandana. He even has a spare for laundry day. In one arc the girls try to get it off of him, and when Jamie finally grabs it off she finds another bandana. But under that they discover his big secret—he's almost completely bald.
Dina of It's Walky! is never seen without her trademark safari hat, even when sleeping or suspended upside down. In the entire series, she takes it off exactly once: when she is about to die.
Rudy wore his cap even when hired by a theme park to live as a wild animal; i.e., naked.
Likewise, Coney kept her bonnet on. In fact, the first time we ever saw her without it (after which she never wore it again) was many years into the comic. Rudy and Lindesfarne must not have seen her without it before either, because they were surprised by her Cousin Itt hair underneath.
Black Hat Guy from XKCD. The only time he was bareheaded was when his hat was stolen (for about 2 strips).
Mr. Mighty of Everyday Heroes always keeps his mask on, even when relaxing at home.
In Girl Genius, the Jägermonsters are almost never seen without a hat. In fact, hats are seen as objects of reverence, and a Jäger without a hat is hardly a Jäger at all.
Choi Dal Dal from Girls Of The Wild never takes off her cat ears no matter what. They are detachable head accessories, though, as show early in the comic as an extra info.
Lee Mi Nam from the same Webtoon is also never seen without his trademark cap.
Web Original
In his show that shares his name, The Nostalgia Critic never takes off his hat. A bit of Real Life Writes the Plot, too: Doug Walker is prematurely balding. Most of his "modern movie reviews" usually have him wearing the Critic hat, too, as some people find his lack of hair distracting.
The entire cast of Red vs. Blue is never seen without their helmets. In the last couple of seasons, most members of Project Freelancer are seen helmetless in flashbacks (excluding Agent Maine, who becomes the Meta), but the guys from Blood Gultch still never take theirs off.
And in an interesting variation, the Director who is also the original Leonard Church is always standing with his face conveniantly just out of frame.
Western Animation
Codename Kids Next Door: Lenny of The Delightful Children From Down The Lane pretty much always wears a football helmet. He only takes it off in one episode where it turns out he's a double agent. Said helmet grows back after revealing he's a triple agent.
Edd from Ed Edd N Eddy. He actually gets very concerned about being seen without it, and when it actually comes off (which is never seen by the audience), the others seem shocked by what's underneath.
Orko never removes his hat, as part of the Trollan custom that they never show their faces to anyone except their one true love. When he and Dree Elle showed each other their faces, they were kept offscreen; only shadows were shown. In one comic book story, Skeletor magically disguised himself as Orko and blew his cover by taking off his hat. Even that was kept in shadow!
The CGI kid racers in The BBC's pre-school show Kerwhizz always wear armoured tracksuits with helmets.
Kick Buttowski never takes off his helmet. He loses it in one episode (his head is covered with mud) and tries to find a replacement.
My Little Pony: Friendship Is Magic has Applejack, who always seems to wear a cowboy hat — even when in a formal dress. Though to be fair, the hat itself is "dressed up", and probably isn't the plain brown Stetson she usually wears. It's also a common meme among bronies to ask where her hat is (since this is My Little Pony, many characters come with their own toys, and Applejack's often comes without her hat.) Still, she does take it off when sleeping, like in "Look Before You Sleep" and "The Cutie Pox," and in one episode she actually took it off in the daytime.
The four main kids of South Park almost never take off their hats. In Kyle's case, it's to hide a really embarrasing Jewfro. Kenny never took off his hood, so no one knew what he looked like until he finally took it off in the movie.
Meg and Chris are almost never seen hatless on Family Guy. Meg is especially notable - going hatless appears to be akin to nudity for her (she quickly covers up when discovered brushing her hair, and a makeout fantasy has her inexplicably hatless). Lampshaded in the Fourth Wall Mail Slot episode when a genie offers wishes:
In Buzz Lightyear of Star Command, Buzz always has a purple helmet liner over his head that completely obscures his hair (or possible lack thereof), even in civilian clothes.
Finn in Adventure Time always wears his hat. He didn't take it off until the second season, for a gag that revealed his long, blond hair. He has taken it off a few times since then, but he definitely prefers life with it on.
Ralphie in The Magic School Bus is always wearing a backwards baseball cap, only taking it off once or twice while sweating, or if the situation calls for him to wear a different piece of headgear (and even then he may still wear the hat underneath). He didn't even take off his hat when he got sick and confined to bed. Made even more bizarre by the fact that he's in an elementary school, which usually has rules against this sort of thing.
Bob's Burgers, Louise never takes off her pink bunny ears hat. She refuses to take it off during gym class. When she was a baby she wore a pink beanie. When her hat was taken away in "Ear-sy Rider," she wore a hoodie to cover her head.
Real Life
Truth in Television: Some religions and cultures require their adherents to wear certain headgear in some situations.
A kippah (yarmulke) is a slightly rounded brimless skullcap worn by many Jews while praying, eating, reciting blessings, or studying Jewish religious texts, and at all times by some Jewish men. In Orthodox communities, only men wear kippot; in non-Orthodox communities, some women also wear kippot. Kippot range in size from a small round beanie that covers only the back of the head to a large, snug cap that covers the whole crown.
Hasidic male Jews tend to always wear black hats in public.
The keffiyeh/kufiya, also known as a ghutrah, mashadah, shemagh or (in Persian) chafiye, (in Kurdish) cemedani and (in Hebrew) kaffiyah, is a traditional Arab headdress fashioned from a square, usually cotton, scarf. It is typically worn by Arab men, as well as some Kurds.
Sikh men wear their turbans whenever they're in public.
Subverted by Terry Pratchett. He is well known for his penchant for wearing large, black fedora hats, as seen on the inside back covers of most of his books, but he actually takes them off now and then. It's just that when he does, people assume that it's not Pratchett. Perhaps it's a reference to his Discworld series? A wizard without his hat is naked. A wizard with his hat is not naked, even when he is.
Actor Mike O'Malley, from Yes Dear and Glee. Just try to find a picture of him without a hat.
Singer John McCrea of the band CAKE is never seen without some form of headwear. He appears to have an affinity for baseball caps.
Defied by many mothers and teachers. When their children or students won't take their hats off, they make them.
