Falls into two major categories, though other variants are possible.
Version 1: Alice, for whatever reason, hands over a possession to Bob and asks Bob not to give it back until X happens. Example: "No matter how much I beg, don't let me get on TV Tropes until I've finished my homework so I'm not distracted."
Version 2: A variant of Self Restraint. Alice allows herself to be tied up or imprisoned and instructs Bob not to let her loose—usually because she is about to be Not Herself in some way, or acquire some dangerous power.
Either way, Alice is doing something she knows is for the best, but she also knows that she'll want out later and needs someone else to keep her in line. She will always tell Bob, "Remember, you can't give in, no matter how much I beg you." Later, Alice will almost invariably beg Bob to give the object back/ let her out of prison. Often she will be entirely justified in doing so because of some development she hadn't foreseen.
There is an exception:
Example 3:(Keep in mind this example is only for non-malicious items.) Alice has left Item X in location D (Her house J miles away) because she does not want it for V reason. Bob finds it, and tries to return it. Alice refuses, but Bob is persistent. Alice says Bob can keep Item X, and Bob doesn't want to take it from her because he finds it "stealing" (or any other reason really.) Alice uses this trope, and Bob walks away. He probably does something with it Alice wants to do, and she asks for it back. Bob refuses, because "no matter how much I beg or what have you, don't give X back."
Whether Bob gives in or not is dependent on the genre of the story. It may be played for laughs with Bob taking the directions too literally and refusing to free Alice even when he obviously should, either out of malice or stupidity. If Bob is especially stupid, he may go the opposite route and let Alice out prematurely when she asks him. This may lead to dire consequences if Alice is a danger to those around her, or may be played for laughs with Alice rebuking him for letting her go.
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Anime and Manga
Light says this to L while requesting imprisonment in Death Note. L is more than willing to oblige.
In Earth X Captain America gave the Black Panther the reality-controlling Cosmic Cube after taking it away from the Red Skull. When a mind-controlling teen starts taking over the world, he returns to ask for it back, but the Panther refuses, just as Cap asked. Later, in Universe X, the Panther also refuses Captain Mar-Vell, but hands the Cube over when Mar-Vell reveals that he's been subconsciously using it himself. Oops.
In Venom, the Batman realises he has become addicted to the eponymous drug and asks Alfred to lock him in the Batcave for a month. In a slight subversion, Alfred asks to let him out early, but Batman refuses. When he finally emerges, he's wearing the Beard of Sorrow, but is, nevertheless, magnificent.
Comedic example: In Young Frankenstein, Frederick locks himself in a room with his newly re-captured Monster in order to counsel it and gain its trust. He gives his assistants explicit instructions not to interfere or let him out of the room, no matter how much he begs them to. It takes the doctor all of three seconds to change his mind once the Monster wakes up and starts approaching him.
"Let me out, let me out of here. Get me the hell out of here... What's the matter with you people?! I was joking! Don't you know a joke when you hear one?! HA-HA-HA-HA! (pounds on the door) Jesus Christ, get me out of here! Open this goddam door or I'll kick your rotten heads in!! MOMMY!!"
By Revenge of the Pink Panther, Clouseau's instructions to manservant Cato to continually surprise him with martial arts attacks, no matter what else he may be told, have become quite frustrating for Clouseau because Cato is now doing it all the time.
Possibly the oddest version happens in Fight Club, where the protagonist's other personality sets this up. Tyler tells his moles at the police department that if he (the protagonist) shows up later to tell them about Project Mayhem, they should cut his balls off.
Though, this is not specific to HIS character. Earlier on in the film they attack another man who was holding a press conference about Fight Club, and threatened to do the same thing to him.
In X2: X-Men United, General Stryker tells his men to kill anyone approaching the room where a brainwashed Professor X is using his mental powers to find and kill all the mutants.
Mystique is in play, so he has good reason for that order— Not that it matters, as Mystique doesn't even come near the room before Magneto triggers the guards' grenades. It's the thought that counts...
Happens in Tropic Thunder, after drug addict Portnoy is informed they are close to a heroin factory and asks to get tied to a tree. His cries to get untied are ignored by the group.
In The Sandlot 2, "The Retriever" attempts to retrieve the NASA scale rocket from the guard dog's yard. He tells the kids no matter how much he is begging to escape, do not let him out. He fails miserably.
Candy doesn't include the scene where Dan and Candy told Casper to not give them drugs no matter how much they begged, but they reference it in the rambling answering machine message they leave him where they're begging him for the drugs.
The Silver Chair: Inverted. The Prince has to be tied to a chair because he's about to have an episode of madness, and he orders Puddleglum and the kids not to free him no matter what he says. Once tied up, he begs to be released in the name of Aslan. The protagonists let him loose, and it turns out that he was Not Himselfbefore; at the same hour every day when he is tied to the chair, he is in his right mind, and letting him go broke the spell.
Older Than Feudalism: In Homer's The Odyssey, Odysseus asks his men to tie him to the mast so he can hear the Sirens but can't escape to go to them. He begs to be released while he can hear the music, but his men have their ears stopped with wax to keep from hearing the Sirens, so they ignore him.
Almost the exact same thing happens in the second Percy Jackson and the Olympians, but Annabeth still has her knife, so she cuts the ropes, and Percy has to go after her. He realizes she can't hear them underwater, so he makes a giant bubble to keep them under and let her breathe.
Harry Potter and the Half-Blood Prince: Dumbledore entreats Harry to make him drink all the potion in the Horcrux cave, no matter how much he begs to stop. Turns into Nightmare Fuel when Dumbledore begs Harry to kill him so as to not endure the mental anguish the potion causes.
A more serious take on this occurs in the sixth book of Animorphs. Jake has been taken over by an alien invader that has complete control over his body and full access to his memories, and the rest of the group decide to tie him up until it starves to death.
In the Star Trek: New Frontier book Stone and Anvil, Captains Picard and Calhoun have their AIs (Data and Morgan Lefler, respectively) refuse to let them issue orders when approached by a ship crewed by a race they suspect can telepathically convince any creature that they want to go along with them.
In The Phantom of the Opera, Christine eventually realizes that Erik is a dangerous, manipulative stalker who has an unhealthy amount of control over her. Even though she knows she shouldn't love him, she does. So she orders Raoul to save her from herself and take her to some foreign country where Erik will never find her no matter how much she resists or protests later. Erik doesn't give them the chance anyway.
According to his friend (and fellow druggie) Thomas de Quincy, Samuel Taylor Coleridge would regularly decide to quit taking opium and hire men to stop him from going into the pharmacist's to buy the stuff. (At the time of course, opium was legal in Englandand mandatory in China.) Unfortunately, Coleridge couldn't figure out a way to stop his future, drug-craving self from simply paying the men some more money to let him into the pharmacist.
A politician in one story by Ephraim Kishon. He lets his wife bind him to a chair, to avoid him going to an interview which would probably uncover his corruption, but he's starving for the publicity and can't resist.
Live Action TV
Friends has Monica tell Joey not to let her open any more of her wedding presents until Chandler gets home. Five seconds later she asks for another one. In a subversion Joey immediately complies. He's no fool; she's a proven obsessive-compulsive with freakish upper-body strength.
In another episode, Chandler tells Pheobe and Rachel he's thinking of calling a girl he's split up with. They think it's a good idea.
Chandler: No! That was a test! In a few hours I'm really gonna want to call her, and you guys have to talk me out of it!
And the one where Pheobe charged Monica with stopping her calling Mike, because they both know there's no future in the relationship. In the end, Mike's impassioned speech impresses Monica so much she encourages them to get back together, much to the disgust of the guyMikehad asked to keep them apart.
ALF had a pretty good example of version 2 in its first season. Every 75 years, all Melmacians go through a day of unpredictable madness and for everyone's safety have to be locked in a cage for the duration. Before being locked up, Alf warned the Tanners that he'd say anything in order to get them to release him, and then spent much of the episode doing exactly that, though in his case he used guile and trickery rather than pleading — at one point even imitating the kids' voices to perfection to make it seem they were locked in there with him. Eventually, he managed to fool Brian into thinking his madness was over, even though the day wasn't, by claiming that the days were shorter on Melmac.
In an episode of That's My Bush!, Dubya accidentally takes some ecstasy pills (thinking they were aspirin) and instructs Princess to lock him in a bedroom and not let him out no matter what so he won't be able to mess up the anti-drug conference taking place at the White House. Unfortunately, she ends up letting him out. That's right. She's dumb enough to be tricked by a high President.
Dubya did display a tiny bit of Genre Savvy before the drugs took effect, though. As soon as The Ditz closed the door, Dubya started screaming "HELP! PRINCESS! OH GOD IT'S HORRIBLE!". She immediately opens the door and comes face to face with a smug looking Dubya saying "No...that's what I told you NOT to do!"
Variation on 3rd Rock from the Sun. Harry tells his bartender boss (this was when he worked in the town bar) to insist that he stay and work if he asks to leave with Vicki. Vicki comes up and Harry asks to leave. The bartender complies with Harry's plan, but Harry becomes more insistent, eventually launching into a rant starting with "Well thanks a lot, you son of a bitch!"
In the episode of Star Trek: The Next Generation "Clues" the entire crew, apart from Data, had their memory erased by aliens who didn't want their existence known. Picard orders Data never to tell the rest of the crew what had happened. It doesn't quite work out as planned due to several pieces of evidence that show that they were missing a day, so they end up going back to the aliens' region of space. The end up convincing the aliens to erase their memories again only this time they are more careful about not leaving any evidence of the missing time.
Which leads to a massive case of Fridge Logic: all it takes is the Enterprise to go to a starbase for regular maintenance and realizing their clocks don't match up for the cycle to begin all over again.
Not positive, but pretty sure the episode techno-babbled its way out of that by saying the Negative Space Wedgie of the day was capable of displacing the ship through both time and space. As such, the clocks being out of sync didn't mean anything; evidence that they had been active for a missing day was the key.
Not to mention why once the Captain contradicted his previous order (which did not include a "No Matter How Much I Beg" Clause), Data didn't simply reveal the truth. Especially because he knew doing so would prevent the crew from facing the very danger they had sought to avoid by his keeping the existence of the hostile aliens a secret.
Because "There is a thing I can't say you, but please don't go in that sector of the space and don't ask why" is not going to work on Picard.
"Genesis of the Daleks" - aversion. The Doctor holds Davros' life in his hands as he forces Davros to order the destruction of the Dalek program. The Doctor then tells Davros to announce that his order cannot be countermanded - but before Davros can relay the order, his PA comes in and saves the day for him. Davros quickly shouts, "My last order is cancelled, repeat, cancelled. No action is to be taken."
Also, in the Series 6 episode "The Girl Who Waited", Future!Amy tells Rory if he really loves her, he can't let her into the TARDIS, even though staying outside would mean her death.
This happened as a matter of course in early seasons Buffy the Vampire Slayer, as Oz's lycanthropy turned him into a Cold Blooded Jelly Doughnut. Further, a similar situation crops up for Xander in The Pack, when he is imbued via Primal ritual with the spiritual essence of a psychotic hyena.
In the seventh and last season, Spike eventually realizes The First Evil is taking over his mind and causing him to eat people again despite his now having a soul. Early in the episode "Never Leave Me" he orders Buffy to tie him to a chair to keep others from harm: "I get free, someone's gonna die."
In the last episode of Coupling, Susan wants to have a natural childbirth, and tells Steve that she's not to be given an epidural unless she asks for it three times; his very first "are you sure?" gets him thrown clear through the delivery room doors.
True Blood, when a necromancer is attempting to make the vampires 'meet the sun'. The restraints of thick silver chains burn through their skin and must be kept there until sunset.
Hannah Montana has an episode where Miley gives Lilly her checkbook for her new bank-account and tells her not to give it back. Leads to a hilarious "Miley doesn't live here anymore" moment when Lilly doesn't give it back.
On How I Met Your Mother, Robin, reeling over a breakup, warns friend (and ex-boyfriend) Ted that some time in the future she will attempt rebound sex with him, and makes him promise to resist her advances. By the time she makes her move, she's such an emotional wreck that Ted finds resisting very easy.
Wings: When Brian and Alex decide to date exclusively, Brian gets nervous about being able to live up to the commitment. So when Joe throws a party at the house for a bunch of gorgeous girls, Brian instructs him to lock him in his room until morning. After he does so, a girl walks out of the bathroom wearing only a towel.
On 30 Rock, Tracy is about to go to sleep and warns Liz not to wake him up, because "I will attack you". Instantly after he loses consciousness he cries out to Liz to save him from his nightmare.
In one episode of Malcolm in the Middle, Malcolm basically cocoons Reese in rope to stop him getting in trouble before his driving test. When they're done, Reese says he needs to pee, to which he gets a response that amounts to "too bad, should have gone earlier." And then Reese congratulates Malcolm on following his instructions.
The episode "Mrs. California" of The Office essentially has its entire "A" plot dedicated to this trope. Robert California tells Andy that his wife wants a job at the Dunder-Mifflin office, but must not be hired under any circumstances. As soon as his wife is in the room, Robert California is insisting and begging that she be given a job. Hilarity Ensues.
When Charlie attempts to quit heroin on LOST (knowing that sooner or later he's going to run out, anyway), Locke takes his drugs and promises to only give them back if he asks for them three times. In two moments of weakness, Charlie demands his drugs back, but Locke refuses to return them. Finally, at the end of the episode, Charlie breaks down and asks for them the third time. Locke hands them over, and Charlie throws them into the fire.
In Peep Show, lovable stoner Super Hans demands that Mark refuse to give him any drugs, even if he turns up at the door brandishing a block of wood. A subversion, in that when Super Hans actually does turn up at the door brandishing a block of wood, Mark hands over the drugs immediately and without further comment.
In the Monk episode "Mr. Monk Visits a Farm", Monk catches Jimmy Belmont setting his illegal marijuana crop on fire. He runs away as the smoke gets very strong. We cut to Randy's farmhand Oates finding Monk handcuffing himself to a tractor as he believes he has inhaled reefer. He appears to have a full panic attack, even saying the trope page quote here, but then instantly sobers up when the sprinklers come on and drench him, prompting him to have the Eureka Moment that solves the case.
I'm going in there, and I'm going to break this man. So no matter what you hear, whatever screams of agony or perhaps pleasure you may hear, I want you to promise me you will not go in this door.
Happened in the comic strip Bloom County when Steve Dallas enlisted Opus' help in quitting cigarettes. He lasted 38 minutes before saying "give me a %$#*& cigarette before I stick you in a blender." Opus refuses, and Dallas quickly goes Axe Crazy on him.
Calvin and Hobbes: Calvin asks Hobbes to tie him up so he can escape from the ropes a la Houdini. But he can't quite get himself free, and then his parents call him for dinner...
When Linus from Peanuts tries to kick his security blanket habit, he literally invokes this trope as he gives the blanket to Snoopy.
Actually, this happened at least twice. The first time, Linus handed his blanket to Charlie Brown, who gave in the moment Linus begged to have his blanket back. Snoopy, however, did not give in at all - thereby causing Linus to have a nervous breakdown.
Although that time he actually kicked the habit - just in time for someone to buy him a new blanket and start the cycle over.
Prior to several of his performances, Steve Martin would tell the spotlight guy to never give him a blue spot, no matter how many times he might ask for one later. Naturally, one of his routines entailed asking for a blue spot and getting increasingly indignant when it failed to show up.
In the Diablo backstory, Tal Rasha gives his followers one of these orders after inserting Baal's soulstone into his own body. Unfortunately, no one told Marius.
In Skin Horse, Tip gives this order upon being possibly infected with lycanthropy. But:
Tip: When night falls, we'll know if I've been infected. Until then...no matter how much I plead, no matter what I may say, do not open this door.
Tip: Wait. There's no bathroom in here.
Unity: Nice try, monster!
Dojo in Xiaolin Showdown makes himself a special prison because every thousand years or so he grows a second head and becomes totally evil for a day. He then instructs the monks not to let him free no matter what until the next day. Omi, of course, eventually falls for one of his tricks.
Rugrats: After raiding the cookie jar gives her a stomach ache from Hell, Angelica orders the babies to stop her from eating cookies ever again, "Even if I beg, even if I cry, even if I threaten to beat your baby brains in!" Hilarity predictably ensues.
Besides the above page quote, there's the episode of SpongeBob SquarePants where Mr. Krabs gives SpongeBob his wallet and instructs him not to give it to him while he's on his date with Mrs. Puff. Before the date even starts, Krabs is begging him to buy something. When SpongeBob gives in and brings it, Krabs berates him for spending all his money. It goes back and forth for a while until SpongeBob snaps, gives Mr. Krabs a manic tirade, and leaves.
And the trope namer is the episode "The Paper," in which Squidward throws away a gum wrapper, and Spongebob takes it and asks Squidward if he wants it back. Squidward gets fed up after a while and tells Spongebob the quote at the top. Cue Spongebob doing all sorts of weird and amazing things with said paper and Squidward eventually trading away everything he owns to get the paper back. Only to find out that he can't do anything Spongebob did with it.
In Captain Planet, when Gaia and Doctor Blight are in a Body Swap, Blight apparently forgot to tell her assistant her Evil Plan, because before switching back, Gaia instructs him to do a bunch of eco-friendly things, no matter how much she tells him to stop later.
On Legion of Super Heroes, in "Brain Drain", Brainiac 5 locks himself in a containment chamber and makes this request when concerned that an interface with other Coluans will make him dangerously unstable. Which it does, but when things go wrong the Legion breaks him out of the chamber anyway. But that all takes a backseat to the Find the Cure plot.
Another version occurs later when Brainiac 1.0 starts to influence his actions. He asks Superman to destroy him if he goes crazy.
Looney Tunes had this one; it was one of the episodes where Sylvester mistakes a baby kangaroo for a giant mouse. At one point asks to be locked in a box with it so he can finally catch it, giving instructions to his son not to open it so the kangaroo doesn't escape. Naturally, he gets beaten senseless, while his son admires his dad's dedication.
Sylvester and his son once took upon a job chasing mice inside a ship. Trying to get out of catching the "giant mouse", Sylvester said that, since there's only one of it and several normal-sized mice, Junior should handle the giant one. After Junior tricked the "giant mouse" back into its cage, Sylvester led Junior outside the room and told him not to open the door no matter what he hears because Junior might dislike what he sees. Soon after that, a mouse Sylvester was chasing got inside a crate full of vitamins made to enhance mice muscles, and continued to pummel Sylvester inside while his son kept his promise and refused to open the door.
In an episode of King of the Hill, Dale asks to be locked in a small vent with a raccoon so that he can capture/kill it. Dale orders Hank and the others not to let him out until it is subdued, no matter how much he begs and screams. Naturally, once he's locked in there, he begins begging and screaming to be let out. Hank lets him (and the raccoon) out, and Dale chides him with, "I give you one little thing to do, and you screw it up."
Done multiple times within the space of one episode of Garfield and Friends with Jon Arbuckle's Little Black Book.
The Mask: Not wanting his alter ego to become Mayor, Stanley instructed Milo to hide the mask and not tell him where it was until the election was over. Stanley would later learn Pretorius was the opposing candidate. Can you believe Stanley would try this again in a later episode?
In an episode of the 2007 version of George of the Jungle, the gang convinces George to go back on his vow to always help people because of him doing menial tasks for Big Mitch. But when Mitch has them captured, they try to call George for help, but George refuses to help on the grounds of what they said.