Kleptomania, from the Greek root words "kleptein" (to steal) and "mania" (obsession with, madness). In the real world, it's an obsession with collecting or hoarding things (which are typically small items like paperclips or pens, which the sufferer may or may not be even aware they stole) thought to be related to OCD. While it might seem quite mild, it can lead to compulsive shoplifting and often appears alongside personality disorders, which can make the consequences (should it lead to shoplifting) even worse.
In fiction, it's the trait of a Loveable Rogue and is frequently played for laughs. "Sufferers" tend to outright enjoy the act of theft, steal anything that isn't nailed down (particularly if it's valuable, in contrast to Real Life sufferers), gleefully enjoy the material rewards and may well be a Karma Houdini for this. They are distinguished from other thief related tropes by the fact that they steal for the pleasure of stealing and frequently have trouble leaving something valuable be. They may or may not be outright described as suffering from kleptomania. Sympathetic characters are often Mr. Vice Guy.
In the animal realm, this is a trait of the packrat and the magpie.
Compare For the Evulz. Characters who might have this trait include the Lovable Rogue, the Gentleman Thief (particularly if they do it out of boredom), and the Token Evil Teammate. They are often an Impossible Thief. The Kleptomaniac Hero is given this trait by the player's own actions (which may be lampshaded as this trope) but not necessarily characterized as such. This character may make liberal use of their Five-Finger Discount card.
You know they've met their match when you hear the words "Oi - Give Me Back My Wallet!"
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Sasha Blouse from Attack on Titan is a Big Eater prone to stealing food. While normally Played for Laughs, her obsession with food resulted from surviving a severe food shortage during her childhood. As such, she tends to habitually snag food whenever she sees it and is affectionately teased by her comrades for her thieving habits.
Tabitha "Boom-Boom" Smith, in Warren Ellis's Nextwave "possesses the mutant powers of blowing things up and stealing all your stuff."
"Fingers", a stage magician and thief encountered by Lucky Luke once. Not only will he constantly pickpocket everyone around him (which means he cannot be kept in prison, because he will casually disarm the guards and steal their keys), but he will regularly offer you your wallet, gun and underwear back, as a gesture of goodwill.
From the film The Thief and the Cobbler, who else but the Thief? He steals pretty much everything he can possibly get his hands on, and tries to steal several things he shouldn't even try to get his hands on. At the end of the film, he also steals all the letters from the words "The End", and then goes on to steal the film-strip itself.
Abu in Disney's Aladdin, in contrast to Al, who only stole as a necessity. This often got him and the others in trouble.
In Tangled, Flynn lifts the satchel with their loot from his own partners in the opening — and at the conclusion, lifts Rapunzel's tiara — giving it back after a dirty look.
Film -Live Action
Allison from The Breakfast Club, who steals the lock to Bender's locker, his knife, Brian's wallet and a patch from Andrew's jacket.
There is a character from the movie If Its Tuesday This Must Be Belgium who starts the trip with an empty suitcase. By the end of the film, the suitcase is full of stuff he's pilfered, including a life preserver from a cruise boat.
In the second Home Alone film, Marv engages in some petty theft with literal sticky fingers: covering one of his hands in tape and then sticking it into a Salvation Army bucket full of change as he walks by.
In X-Men: Days of Future Past, Quicksilver's basement is full of stolen goods such as several TVs, a Pong arcade machine and an entire shelf of junk food. His mother, upon opening the door, assumes Wolverine and co. are cops and simply offers to cut a check for whatever he stole.
Moist von Lipwig- it's a Running Gag that he keeps stealing Drumknott's pencil and certain samples of paper.
And Nobby Nobbs is a fellow kleptomaniac. Thud! brings up the problem of why Nobby is allowed to be on the City Watch despite his thieving ways. It's because he's an old friend of Sam Vimes, he seldom steals anything of value, and (like Fred Colon) has a keen ability to read the mood on the streets as well as his ability to sneak around where the (worse) criminals can't see him. Nobby's ability to steal without being seen has also come in handy a few times, and is part of the reason he is still employed.
Mulch the kleptomaniac dwarf from the Artemis Fowl series fits this trope. He also likes to steal things just for the challenge.
The main character of Mary Anderson's book, Step On a Crack suffers from an odd sort of kleptomania where she feels compelled to steal certain items over and over in conjunction with nightmares and occasional fugue states. All of her problems stem from repressed memories of her real mother.
The eponymous Locke Lamora, of The Lies of Locke Lamora, once had it said about him that "if he had a bloody gash across his throat and a physiker was trying to sew it up, Lamora would steal the needle and thread and die laughing. He... steals too much." Lamora was 5 years old at the time.
Magnet from Holes. Things just seem to stick to him wherever he goes. Unfortunately, a lot of those items don't belong to him, so he wound up at the juvenile camp.
Played for Drama (a small amount of drama) in the X-Wing Series novel Wraith Squadron. A bit character who interviews to join the eponymous squadron is rejected when he, apparently compulsorily, swipes a framed picture from the commander's desk. (It's mentioned that he is facing charges for several other, equally petty, thefts, but is confident he'll be cleared.)
Live Action TV
Played for Drama (and somewhat more realistically) in an episode of Law & Order: Criminal Intent when a couple of murderers are caught when one of them can't resist stealing an eggcup from their victims' home. This leads to a somewhat Narmy ending where her partner screams "EGGCUP!" at her while he's hauled off.
Parker from Leverage. Interestingly she actually poses as a kleptomaniac when the team needs to infiltrate a rehab facility.
Mike Hamar of The Red Green Show is a robber on parole who hasn't had an honest day's work in his life. A Running Gag is that he frequently has to return small items to the people he's talking to, having picked their pockets out of force of habit.
A more realistic than usual example on My Name Is Earl of all things, with a recurring character who is shamefully compelled to steal pens and only pens, regardless of how many he has already, and regardless of the fact that he doesn't even want the pens. Still Played for Laughs, though.
The titular character of Sherlock. He breaks his flatmate's computer password to use the laptop ("Mine was in the bedroom"), has stolen more than one of Lestrade's police IDs (he comments that he "pickpocket(s) him when he annoys me"), and has taken the precaution of taking his older brother's ID in case he might ever need it.
Amanda from Highlander. She's more or less the Hollywood version, with a taste for expensive stuff.
An episode of Alf dealt with this when one of Tanner's neighbors is revealed to be a kleptomaniac. Alf caught her stealing some of the family's jewelery and confronted her son, who he had become friends with, and he had to explain that his mother simply couldn't help it.
In Tricia's back-story in Orange Is The New Black, she stole even while trying to pay a store back for something else she stole.
In Help!!, George Harrison is seen sneaking various items into his pockets as the Fab Four see the local jeweler.
The Exodus song "Force Of Habit" is about someone fitting this trope.
The Kender from the DragonlanceDungeons & Dragons setting are convinced that they're only borrowing interesting things in good faith, and will certainly return them. Except they often wander off and forget the whole thing... but they're utterly convinced that they're not doing anything wrong. In fact, accuse a kender of being a thief and he will protest at length. Kender always have dozens of pockets in which they keep their various pilfered goods. As a gameplay mechanic, a Kender can check their pockets to try and find an item of trivial value that they have stolen without remembering.
Tabletop Game/Warhammer40000 has the Blood Ravens, who are also known as the Blood Magpies due to their tradition of grabbing ANYTHING that might possibly be a clue to their lost history, a useful weapon, bling or for the heck of it.
You! Yes, you, in countless video games, especially Role-Playing Games. Pretty much anything that's not part of the background is potentially yours. Affects moral alignment in games that have such.
Shir Gold from Phantasy Star II has a random chance of stealing from every armory, armor shop, or item shop you enter with her in the party.
Garrett of Thief usually gives "the rent is due" as an excuse for his thieving, or has some other objective he needs to accomplish. However, it is clear from his actions and dialogue that he really enjoys theft and is, at least partially, Doing It for the Art. For example, when a museum in Deadly Shadows boasts that its security is impenetrable, Garrett takes it as a personal challenge.
Billiken is a more aggressive version of this in Devil Survivor 2 - he compulsively mugs other demons (or tamers) and has no qualms about killing. Once he depletes your wallet, he can hit you with the powerful Barrage Strike move.
The twin ghosts at the Labyrinth of Amala in Shin Megami Tensei III: Nocturne, on the other hand, will take the nonviolent approach: they will offer you A Spot Of Tea for a very low sum. The tea is drugged, and they will kick you and our demons back into the entrance of that Kalpa, after helping themselves to more of your money, of course.
100% Orange Juice: Marie Poppo kickstarts the plots of the various campaigns by stealing other characters’ valuables. This is also demonstrated by her Hyper “Ubiquitous” where it has her teleport to another character and steal their stars.
Ultra Fast Pony has Applejack stealing random stuff, off-screen. Like trees, bicycles, and medical supplies.
Sam Starfall in Freefall is good enough at pickpocketing that his sticky fingers become a problem when he tries to go honest.
Thief in 8-Bit Theater, who steals from everyone (both via outright sleight of hand and contracts with legal mumbo jumbo) to finance the search for a cure for his sick father. When this is done he still steals everything that's isn't both nailed down and on fire.
The pixie Feiht in Chasing the Sunset regards anything shiny as there for the taking. (Spelling her name backwards gives you a clue to her personality.)
"The Raccoon" from Kiwi Blitz. He steals "random crap" purely for thrills, and doesn't mind returning it when forced to do so. A later flashback shows that he literally has kleptomania.
Megatokyo: Meimi Sonoda and her daughter Yuki. They often only realize they have stolen something when on their way home. Their kleptomania is tied to their powers. Meimi is an expy of Saint Tail.
Sette Frummagem from Unsounded. Her dad is a head of the thieves' guild, so it's no surprise that she would become a pickpocket. What is a surprise is how good she is at it.
Sidney Malik Of Widdershins is cursed with this. Walking though a crowd could result in a extra ten wallets with no effort from him. He isn't happy with the curse and it may have played a part in him being expelled.
Fry: Come on, Bender. I'm sure there's plenty of cool stuff to swipe. Bender: I don't know, Fry. I feel like, for the first time in my life, I've stolen enough. Fry: What?!? Snap out of it! (slaps Bender) Bender: Wow, thanks buddy. Don't know what came over me. (sing-song) Oh, I love stealin'. I love takin' things.
Sonic's brother Manic has this problem. It stems from his upbringing as a thief's adopted son.
Said adoptive father might also count, given the first thing we see him do is steal Manic as a Doorstop Baby (left on someone else's doorstep).
Mr. Krabs in SpongeBob SquarePants. One episode had an amusement park hosting a "Free Day" where everything in the park was free. Mr. Krabs took this to mean everything, including streetlight lightbulbs and the egg of the star attraction.
Homer Simpson from The Simpsons has been known to steal beer mugs from Moe's, office supplies (including computers) from work and especially just about anything from Ned Flanders. One episode had him get a job at a Wal-mart parody only to find out every employee has a shock collar on their neck. When his coworkers told him they all knew how to disable the collars and steal everything that's not nailed down, they asked him not to judge them. Homer not only doesn't judge them but he steals an entire cart of TVs right then and there.