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Black Butler, especially season 2; Sebastian contractually owns Ciel's soul & literally kills anyone who tries to kill Ciel before he can, even other demons. In the anime he actually does try to murder Ciel.
Gintama - Kamui gets excited watching Gintoki fight Hosen. He declares Gintoki is his (prey) and that he won't allow anyone else to touch him. He even leaves Yoshiwara alone so that nobody will come after Gintoki, and to a lesser extent Kagura.
Goemon and Lupin from Lupin III is an archetypal example of the Heel-Face Turn variant of this trope in Anime. In addition, Detective Zenigata and Lupin are another example from the same series.
Nagi and Ryoko in the 1st Tenchi Muyo! TV series. Nagi joins in the heroes' effort to save Jurai from the usurper Kagato, just to make sure nobody else gets a chance to kill Ryoko.
In a strange inversion in Naruto, Itachi Uchiha basically says that the only one allowed to defeat him is his younger brother, Sasuke. In Itachi's case this is all part of a larger plan: he intentionally died, making Sasuke think he won, to make Sasuke strong enough to survive Madara and redeem the clan's honor. That...didn't exactly go as planned. Also, Naruto and Sasuke, mutually. Madara also intends for Sasuke and Naruto to fight as he wants them to settle an old ideological grudge between the Uchiha and Senju clans, with Naruto symbolically representing the Senju.
Gaara at one point declares a statement like this, marking Sasuke as his prey. It's a subversion as Sasuke is brushed aside rather quickly, not being up to snuff to play with monsters, and Gaara eventually gives up on the idea altogether.
Also, the sound ninja Dosu wants to be the one to fight Sasuke in the Chunin exams, so he tracks down Sasuke's assigned opponent, planning to kill him. Unfortunately that opponent was Gaara, who murders Dosu in about five seconds.
Plus, Raikage wants to be the one who defeats/kills Sasuke. He abandons it after he learns that Killer Bee, his little brother and the guy Sasuke captured, actually trolled Sasuke and the whole village to take a vacation.
Deidara had it out for both Itachi and Orochimaru, for being the reasons he got stuck in Akatsuki in the first place and then Sasuke happened...
In Noir, Chloe decides to go for a walk with her assassination target (who knows perfectly well why she's there). She then kills a group of men that was also sent after him. When he expresses his gratitude, she reminds him that she is still going to kill him — she only saved him because if the other group killed him, she wouldn't be able to fulfill her mission.
GrahamAker of Mobile Suit Gundam 00 believes that he's the only one allowed to defeat Setsuna F. Seiei (or more to the point, Setsuna's Gundam). Setsuna has more important things to do.
In the side stories, this trope is how Fon Spaak's relationship with Celestial Being begins. In 00P, he manages put together some clues and begins to suspect the existence of some sort of secret organization. He then organizes an ambush and has his ass promptly handed to him a Gundam, but Celestial Being's supercomputer orders the pilot to let Fon go, because he's too good to just kill off and won't leak the information anyway because... well, see the page title.
Becomes the main theme throughout Samurai Champloo, especially with the main characters: Jin and Mugen each live this trope in their friendship and ardently believes that they are only ones to kill each other. This also is the only thing that keeps them with Fuu throughout the show as they promised her that they would not kill each other until they help her find "the samurai that smells of sunflowers". Additionally Fuu spends the entire show trying to find the samurai who smells of sunflowers so that she can exact her revenge against him for abandoning her and her mother when Fuu was little. She is robbed of this pleasure when she finds him dying in the last episode.
Also, the secondary and tertiary characters follow this trope with many of Jin's former dojo partners hunting him down so they can exact their revenge against him for killing their master, all while believing that none should rob them of the opportunity to do so. Mugen too was betrayed by his friend in the past and thus thought that he would be the only one to kill him too. Sheesh, this show must believe that revenge is a dish best served by yourself.
Created an interesting situation in which one of Mugen's old partners shows up and subsequently betrays Mugen, seemingly killing him. Jin proceeds to kill him for killing the man he was supposed to defeat, but when Mugen shows up alive, he is extremely angry at Jin for killing the other man he thought only he was allowed to kill.
Hajime Saitou and Kenshin Himura in Rurouni Kenshin. Also, to a lesser extent, Aoshi Shinomori and Kenshin Himura.
And Sanosuke Sagara and Saitou, though that is one-sided on Sanosuke's part.
Kirisawa Fuuko and Hanabishi Recca, Ishijima Domon and Recca, and Kurei and Recca (mutual) in Flame of Recca.
Oogami Souma and Tsubasa in Kannazuki no Miko. In an odd, twisted way, also Himemiya Chikane and Kurusugawa Himeko.
Seto Kaiba and Yugi Muto, or rather "the other Yugi", in Yu-Gi-Oh!. In Season 4 of the anime, Yugi loses to The Dragon. When Kaiba hears this, he throws what can only be described as a "temper tantrum".
In season zero Kaiba defeats the three Hekate for Yuugi as Yuugi's puzzle was stolen and thus could not transform. The final words of the episode were him saying "And don't forget! The one to defeat you... Has to be me."
Bakura also, as he wants Yugi's Millennium Puzzle and thus will allow no one else to claim it before he does.
Yugi Muto and the pharaoh, actually, when it comes time for the Ceremonial Duel. Even in the anime when Kaiba strangles Yugi, Yugi refuses to let anyone else fight his other half. And then Yugi totally wins.
In Yu-Gi-Oh! ZEXAL Gauche desires to be the one to defeat Yuma (because Yuma has what he believes is "the original Number") so much that he allies himself with Yuma when it looks like there is a chance that someone else will beat him to it. Unfortunately for Gauche, while Yuma does lose a couple of duels, he isn't the one who defeats him. (Note that Gauche really isn't a bad person... He's just very competitive.)
Goku and Vegeta in Dragon Ball Z. Also Goku and Piccolo (Junior) early on, which is why they teamed up against Raditz in the first place. After Piccolo got to kill them both, his motive for being a villain dried up - he claimed that the revived Goku would be next if they could beat the Saiyans, but the Heel-Face Turn had already taken hold by that point.
In one of the non-canonical movies, Vegeta shouts "Kakarotto wa ore no mono da!!" ("Kakarot is mine!") when Android 13 is beating his rival to death, and charges in to protect Goku. Obviously in context it's something like "he's my prey, screw off," but it was all too easy for Yaoi Fangirls to squeal "Ho Yay!"
Likewise (but without the Ho Yay potential), when the main series, Goku (weakened by disease) is being trounced by Android 19, Vegeta comes to his rescue. He then informs 19 that "Nobody kills Kakarot while I'm around! Destiny has reserved that pleasure for me!"
Well, that is because of this trope and the fact that Iscariot is a rival vampire-killing organization anyway.
In Afro Samurai, the hero is, literally, the only one allowed to defeat the main villain. In contrast, for the sake of multiple action sequences prior to the climactic battle, just about everyone and his uncle is allowed to challenge the hero. The reason: the main villain has the "Number 1 Headband", while the hero has the "Number 2 Headband". Only the Number Two can challenge the Number One, but anyone can challenge the Number Two.
Based on the final fight between Justice and Afro it would seem in reality that the only one who should've been allowed challenge the Number 2 was the Number 3, Number 10 being the one anyone and everyone could challenge. Given the other 8 headbands had been lost/collected by Justice and nobody knows about them, Afro is the one left dealing with the irritating hoards of wannabes.
Soukou no Strain: Sara and Lottie's initial roadblock to being True Companions is that they both claim to be the only one that can or should defeat Ralph. Lottie eventually relinquishes her claim to Sara.
Tyson and Kai in Beyblade. And Tyson and Ray. And Tyson and Max. And Tyson and Daichi. And...
Ichigo and Zaraki Kenpachi in Bleach. Although, Ichigo wants nothing to do with that fight. Zaraki tends to want to go to the Human World whenever Ichigo's involved with anything there so he can fight him again. Though it's not so much that he wants to kill him, he just wants a sparring partner who can actually challenge him.
Later, Ichigo with Grimmjow, to the point where there is not one rematch, but two. Also, Grimmjow gets inordinately pissed when he finds out other rival Ulquiorra's been going after "his prey."
Loly pulls out the stops for this one. She wants to kill Orihime twice, but when Orihime's threatened by Yammy, even though Loly clearly stands no chance, she calls on her released form to defend the human.
In Kikaider, Hakaider/Saburo's main objective is to destroy the titular android. He resolves the existential crisis of that being his only reason to exist by destroying any other Monster of the Week that comes close to killing Kikaider, while pushing him to his limits so he can get his money's worth out of his goal when the time comes.
Kyuzo embodies this trope with regards to Kambei in Samurai 7.
Variant: Golgo 13 makes a point of not allowing anyone else to kill his target when hired for an assassination. Not even themselves. When they die, it must be by his hand.
InuYasha: When Sesshoumaru initially enters the story he doesn't seem to care who, if anyone, kills Inuyasha, as long as he gets possession of the sword that their father bequeathed to Inuyasha. As the plot goes on, however, his behaviour becomes much more complicated; it becomes increasingly common for him to aggressively enforce his claim on Inuyasha's life whenever a situation occurs that could involve Inuyasha losing his life to others. It becomes increasingly clear it's a face-saving excuse to step in whenever Inuyasha and even any of his friends are in trouble without Sesshoumaru having to admit that he's not Just Passing Through. Eventually, even the excuses stop when, during a fight with the Ultimate Evil, he openly admits he's protecting them all.
In the 2003 anime version, Envy literally bases his entire existence on killing his father/homunculi-creator, Hohenheim of Light. Then he throws a floor-destroying temper tantrum when he thinks Dante beat him to it.
Rukawa and Sakuragi to Sendoh, and later Rukawa to Sawakita in Slam Dunk.
The reason Van and Ray are opposed to each other in GUN×SWORD is because both are determined to be the one who kills the Claw.
Papillon of Busou Renkin takes this to its logical conclusion, being so single-mindedly obsessed with ensuring that nothing and no-one else kills the protagonist Kazuki Mutou that he effectively ends up as one of the heroes' most valuable allies despite technically being a villain.
In Rozen Maiden Träumend, the first season's Big Bad Suigintou gives an entire speech about how she will be the one to ultimately defeat Shinku while taking a fatal barrage of crystal arrows, shielding Shinku from harm and dying in her arms.
Pixy Misa in Magical Project S often talks about how she will be the one to defeat Pretty Sammy and nobody else.
Kyouko, the main character of the manga Skip Beat!, views her relationship with Shou in this way.
This is pretty much Aptom's raison d'etre in Guyver, his sole goal in life is to defeat Sho in combat.
Until their promotion in Pokémon: Best Wishes, the only reason that the Team Rocket trio continued to exist in the show was for the sole purpose of capturing Pikachu, giving it to their boss, and becoming ridiculously wealthy, according to their fantasies of how well their boss will reward them.
Rather explicitly used in Cowboy Bebop with Vicious and Spike, with Vicious even telling Spike in the final episode: "I've told you before that I am the only one who can kill you." Unlike most examples of this trope though, Vicious never rescues Spike from any other opponents who might be able to off him. This line weirdly seems to imply Spike can't be killed by anyone else.
Atobe Keigo's attitude towards Tezuka Kunimitsu in The Prince of Tennis. In the manga it's more like the most prominent of several rivalries that Atobe has with several players (Sanada and Echizen are the other two), while the anime (and the fandom) exaggerates it to the point of almost stalkerish obsession.
The Lucifer and Biscuit Hammer: "Princess" Sami is determined to save Earth from a mage bent on cracking it open with a gigantic hammer... so that she can destroy it herself!
In Baccano!!, while Ladd Russo and Lua Klein are not rivals, Ladd does promise to be the one to kill her, and will let no one else do it.
Youhei: "No! You're supposed to say, 'I don't want to see you lose to anyone other than me.'"
In Battle B-Daman, Anti-Hero Enjyu is obsessed with defeating The Hero Yamato Delgato and ends up secretly helping the main gang because doing otherwise would let Big Bad Marda-B defeat Yamato.
Yami from To Love-Ru acts this way towards Rito, to the point of saving his life repeatedly so that she can kill him. Of course, she's had dozens of opportunities to off him, but never actually does, so it seems that the whole "only I can kill him" thing is just an act to justify saving him all the time.
Or it is because, as Yami justifies it, so that she can stay on Earth since she likes it there.
From Liar Game, it appears that Yokoya is heading towards this path in regards to both Nao and Akiyama. At first, it seemed that he only wanted to crush Nao in an attempt to defeat Akiyama. But after the Pandemic Game, he's got his sights set on defeating Nao as well.
Suzaku is this to Zero (and, indeed, whoever he feels is "wrong") in Code Geass, though somewhat subverted in that he does not go out of his way to protect them from others who decide to take the opportunity. He will complain, but that's about it.
This is pretty much his role in Zero Requiem.
The jury is still out, but this may have been Trafalgar Law's motivation for saving Luffy in One Piece due to him frequently denying considering Luffy a friend.
He says he considers Luffy a Worthy Opponent and thinks it would be "lame" for Luffy to die prematurely.
Monster subverts this in that Johan handpicks Tenma to be the one to kill him, but it is also played straight in that he will not let anyone - the police, random Nazis, or sundry criminals - lay their hands on either Tenma or Nina without aid to the two and swift retribution to their enemies.
Hayato and Kaga in Future GPX Cyber Formula, as Kaga sees his rivalry with Hayato as this, In fact, he doesn't anyone else to be his rival, and he even beats Hayato in his final race in SIN.
In Risky Safety, Risky decides to try and rescue a young girl from burning to death, and prior to that, yells at to not go into her father's burning workshop. Why? Because Risky's supposed to take her soul, and she can't do that if the girl dies by someone/something other than Risky herself. Or so she claims, anyway...
Both played straight and averted in Black Cat. Train and Creed seem to have a mutual attitude along these lines, although neither of them actually go out of their way to help the other. Also, in volume four of the manga, an assassin appears who believes that killing is an art. After witnessing Train shoot bullets out of the air, the assassin ignores his current target and leaves, before flat-out telling Train (paraphrased) "You'd better stay alive, so I can kill you myself." Averted, in that the character never makes another appearance.
Medaka Box: Pretty much invoked word for word by all of Medaka's former enemies, including the Plus in support for her against Kumagawa.
In chapter 213 of D.Gray-Man Kanda tells Link "If your mission is to assassinate bean sprout (Allen), I'll cut you down where you stand. I'm the one who'll kill him."
Sheila of Superior claims, repeatedly, that this is why she won't let anyone else kill Exa. She also claims she won't kill him herself because she's not done toying with him yet. (It becomes obvious pretty quickly that she just likes him too much to want him dead.)
Somewhat inverted in Death Note where LightYagami seems to feel that L ought to be the only one allowed to defeat him, and he's very bent out of shape when somebody else takes over and does it instead. This is also one fan explanation for why Light sees L as he's dying, though there are other theories.
Played with in The Wallflower. Earlier in the manga Sunako states this to be her reason for saving Kyouhei from anyone who tries to hurt or kill him. As the series progresses however, it becomes a very obvious excuse.
In Vinland Saga, main character Thorfinn has sworn a blood oath to kill his raider band's leader, Askeladd, for killing his father and repeatedly (and irrationally) protects his own father's killer from harm in order to be able to kill him in a "fair" fight. It becomes a deconstructed trope as it's made clear that ten years with this attitude has more or less ruined Thorfinn's life and he wouldn't know what to do if he ever killed Askeladd, which he's unable to do anyway because Askeladd pretty taught him how to fight in the first place and can beat Thorfinn effortlessly every time he's challenged. Askeladd eventually calls Thorfinn out on his entire attitude and calls him an idiot. He is stabbed fatally by Canute on the following day and Thorfinn finds himself unable to even give him a Mercy Kill, instead going into a Heroic BSOD and attacking Canute.
In The Devil is a Part-Timer!, this is the default attitude of Emi Yusa, hero of Ente Isla, towards her arch nemesis, Satan, or Maou Sadao as he goes by on Earth. Unfortunately because neither one of them can use their magic for very long and Maou appears to be a regular human due to his lack of magic, to Earthlings she just comes off as a Clingy Jealous Girl and Stalker with a Crush who hasn't gotten over her breakup with Maou.
When Superman was killed by Doomsday, Lex Luthor (disguised as his own son; long story) starts attacking the corpse of Superman-killer Doomsday in a rage with a chair. The people present assume it's because he's angry at Superman's murderer. He is, but not for the reasons they think.
He reacted the same way (and for the same reason) when he thought the Silver Banshee had successfully killed Superman.
The film adaptation of the Doomsday storyline, Superman: Doomsday, has a similar reaction from Lex. He's pretty ticked off that an "intergalactic soccer hooligan" robbed him of the chance to defeat Superman with some sort of brilliant Evil Plan. Of course, Lex was responsible for releasing said hooligan, but even then he can't take credit because Mercy Graves destroyed the evidence. So he kills Mercy instead.
One more for the road: Early on in Post-Crisis history, Superman's first battle with the Kryptonite-powered Metallo went badly for the Man of Steel. Just as the villain was about to finish him, Lex's agents arrived and took Metallo away. In Superman Villains Secret Files, Lex explained the issue to his infant daughter with "Well, I couldn't allow a fool like Corben to enjoy the killing blow, could I?"
The Joker extended this trope to Robin (Tim Drake) at the end of the mini-series Robin: Joker's Wild. After being defeated by Robin while Batman was out of town, Joker sat angrily in his cell at Arkham, warning the other inmates, "No one touches the boy, d'ya hear? He's mine!"
And of course, The Joker to Batman himself. He has, on numerous occasions, proclaimed that his only reason to live is to kill Batman and throws mad rages (or even completely snaps and turns sane) when he thinks somebody else did the job. Furthermore, he proclaims the self-imposed parallel as well where all he wants is to drive Batman to the point where Batman will kill him, thus crossing the line.
Before the New 52 reboot, Joker had taken it upon himself to kill anyone who tried to kill Batman; only he can kill Batman & only Batman can kill him. Nothing more romantic than double homicide & hate is just another kind of love.
Death Of The Family: Interestingly enough, Batman has taken this attitude towards Joker, and is even turning down the Batfamily's offers to assist him. This may come back to haunt Batman....
Batman used this trait as a Batman Gambit in Mad Love to save himself from Harley. Puddin'.
Megatron uses this trope to his advantage in the Transformers: Shattered Glass comics. He knows full well that an Autobot would never kill him out of fear of what Optimus would do to them later for destroying Megatron before he got the chance. They're even afraid to tell Optimus that Megatron MIGHT be dead. For those unfamiliar with Shattered Glass, it's a universe where the Autobots are evil conquerors and the Decepticons are heroic freedom fighters.
This is something of an inversion of their relationship in the MarvelGeneration 1 comic. In a story by Bob Budiansky, Optimus and Megatron agree to settle their differences by video game tournament, and Megatron wins (by cheating). Thus, the ref blows up Optimus, and so Megatron slides into depression and insanity due to not being the one that struck the fatal blow. In fact, he is so obsessed over the matter, that when Brawl tries to console him, Megatron crushes his head; and when Brawl, the most Ax-Crazy of the Combaticons, is trying to be the voice of reason, you know Megs has gone over the edge.
Megatron in Regeneration Onenote the IDW series that continues the original Marvel continuity, splintering from both the Generation 2 and Classics stories is determined to get Optimus this way, even if he has to die himself. He even taunts Prime with what he's done to Earth in the two decades it's been since the Autobots went back to Cybertron.
Daken has this with his father; he fought Deadpool when it seemed he was about to kill Logan. Of course Wolverine set the whole thing up.
Daken: He's my father. And I'm going to kill him for it, not you.
Skaar has this for his father the Hulk and it's why he protects Banner while he waits for the Hulk's return. Granted, Skaar isn't a bad guy (He's mainly angry because he thinks he abandoned him) and Banner is training him for when the Hulk returns.
Doctor Doom is perfectly willing to save the Fantastic Four from certain death at the hands of anyone else, just so that he can kill them himself later. However, he only gets involved if Reed Richards is with them — if only the other three are in danger, he couldn't care less.
In a storyline where the original Fantastic Four were killed and their positions taken up by Spiderman, Wolverine, Hulk and Ghost Rider, Doom stepped in and killed the enemies threatening them, saying "None may defeat the Fantastic Four... save Doom." Evidently being stand-ins count.
This is Dr. Eggman's attitude toward Sonic in Archie Comics' Sonic the Hedgehog. In the 2010 Free Comic Book Day issue, one of the badniks created by the original Eggman/Robotnik is about to kill Sonic...and he proceeds to destroy it so it wouldn't "ruin his time-table", much to Snively'simmense frustration.
In the Marvel Adventures remake of the Iron Man storyline, "Armor Wars", Doctor Doom appears halfway through as a Doombot, allowing Tony to use the armor to aid in his quest to recover his armor. At the end of the story, Tony talks to Doom, revealing that he wasn't going to let some "stale, Cold War leftovers" kill Tony when he would be the one to do so.
Averted with tragicomic consequences in an issue of What If? where The Punisher succeeded in killing Spider-Man during their first meeting. His various villains throw a party and invite the Punisher as the guest of honor. Unfortunately for them, they've misunderstood Frank's motives, and he takes the chance to mow them all down.
In the New 52, the Black Manta has been obsessed with killing Aquaman, ever since Aquaman accidentally killed Manta's father. When Manta thinks Aquaman is dead in the Forever Evil event, he vows revenge against the people who claim to have killed him, the Crime Syndicate.
Black Manta: The Syndicate killed Aquaman. They took the only thing I wanted from me. So I'm going to take everything from them.
In one run of The Punisher, Frank is seemingly executed in an electric chair, only to have his death faked by a mafia family who have their own plans for him. Frank's arch nemesis Jigsaw dresses up like him and goes around killing anyone who had anything to do with the sentence. When he finds out that Frank's alive, he's initially overjoyed. Then he realizes that he killed all those people for no reason, and rages at Frank for making a fool out of him.
Queen Of All Oni: When Ikazuki selects Tohru as his new host, Jade lashes out at him, since she views Tohru as her Arch-Enemy and doesn't want anyone else to harm him. Unfortunately for her, Ikazsuki is more powerful and forces her to do things his way.
Subverted by Slade when the Titans are captured by Mad Mod during the second arc — he states that whether or not the Titans succumb to Mod's brainwashing, it doesn't affect his plans, so he doesn't care.
Firefly from Ace Combat The Equestrian War won't let anypony else, but herself take Black Star down. This is because she promised herself to avenge her parents by defeating their murderer.
A rare heroic variant occurs in the Super Smash Bros. based One Last Smash. In it, the Nintendo Universe is protected by the characters from the first two games. Eventually, someone starts murdering all of their enemies. While most of them are unsettled by this, Donkey Kong is outraged that someone killed K. Rool and his army, feeling that he was denied a final confrontation.
Donkey Kong: Gonna smash guy who smash K. Rool without tellin' Donkey.
Trixie: Besides, Mr. Wrong is Trixie's! He shall feel the GREAT and POWERFUL Trixie's wrath before this trial is over!
In Substitute Harmony, after Pinkie Pie is falsely implicated for the disappearances of her friends, Trixie shows up and wants to kill her... because she wanted to defeat Twilight Sparkle first. Gilda shows up for similar reasons, as (she believes) Pinkie has robbed her of her last chance to patch things up with Dash.
Bass is also this to a much more frightening degree.
In Mega Man Recut, Proto Man is slowly becoming this towards Mega Man.
In Once Upon a Time in the West, Harmonica is the only one allowed to defeat Frank. When several of Frank's own men are paid to kill him (Frank), Harmonica shoots one of them. When he is accused of saving Frank's life, he defends his actions with, "I didn't let them kill him, and that's not the same thing."
In The Purge during the assault on the Sandin family, the family's neighbours enter the house and dispatch their attackers, saving them from violent deaths. However they then reveal that the only reason they intervened was so that they could kill the Sandins themselves as their own Purge targets.
In Die Hard, after John McClane kills Karl's brother, when Karl goes to the roof with two other henchmen, while riding in the elevator, he instructs both of them about John: "No one kills him but me."
Earlier on, Joker had also prevented Coleman Reese from revealing Batman's identity (claiming he didn't want Reese "spoiling everything") by issuing his infamous "Either Reese dies or I blow up a hospital" challenge.
The Neverending Story inverts it: G'mork, the servant of the Nothing, claims that Atreyu is the only one who can defeat him. Saying that wasn't a smart move, because it gives Atreyu the Heroic Resolve to kill him.
In X-Men Origins: Wolverine, as Weapon XI is about to decapitate Logan, he is tackled away by Victor Creed. "Nobody kills you but me!"
In the aftermath of the climatic battle at the end of Gangs of New York, Bill the Butcher is mortally wounded by shrapnel from a random cannon blast. Amsterdam is furious that he wasn't able to kill Bill himself and properly avenge his father.
In Enemy at the Gates, the Nazi sniper tells one of the characters, "He isn't dead. Do you know how I know that? Because I haven't killed him yet." It should be mentioned that the line is rendered extra-creepy by the fact that it's Ed frickin' Harris.
Luke and Vader in Star Wars. Yoda even says "You must confront Vader."
An interesting example occurs earlier in A New Hope when Vader senses his old master, Obi-Wan Kenobi, is aboard the Death Star. When Tarkin suggests they make moves to prevent his escape Vader informs him that "Escape is not his plan" and that he must face him alone. Vader believes his master has come to confront him and desires to finally defeat him. Kenobi is not only aware Vader will come to face him, but is counting on it to ensure Luke escapes whilst Darth Vader's attentions are elsewhere.
In Circus, protagonist Leo incurs the wrath of Moose, who wrongly accuses Leo of murdering his mistress, as well as a pair of Loan Sharks that he owes a serious debt to. This leads to a scene where Leo is being threatened by the Sharks, Moose arrives and beats them up... then announces his intention to kill Leo and chases him down the street. The scene ends with Troy, one of the sharks, recovering from the attack:
Troy: "He can't bloody kill him... I'm gonna be the one to kill him!"
The movie Bullitt has this going on between Tupac's character and Micky Rourke's character.
In the HBO film Gotti, it's suggested that the Feds actually helped John Gotti avoid getting killed by enemy families, just so they could bring him down themselves.
A more reasonable example occurs in Skyfall. Silva's actual grudge is against M, and he tells his men that only he can kill her. He doesn't care who kills Bond. When he sees that M has already been mortally wounded, he practically breaks down.
A curious subversion in The Hobbit. Azog the Defiler has no problem having one of his underlings collect Thorin's head. Presumably he thought he was already dead.
Played with in The Chronicles of Riddick when the Lord Marshal uses every means at his disposal to try and kill Riddick, before deciding to fight him personally. It's a Double Subversion. In the backstory, he heard a prophecy that a Furyan could kill him, so he slaughtered pretty much the entire planet.
Harry Potter - A prophecy made before Harry was born foretold his rivalry with Voldemort, and that one of them will end up killing the other. The exact terms could have referred to someone else (Neville) born the same month, but Voldemort chose to go after Harry first - and created a Self-Fulfilling Prophecy thanks to Lily's sacrifice. Years later, Voldemort remains obsessed with killing Harry personally.
Subverted when Harry returns to Hogwarts in Deathly Hallows when he invokes something like this - Neville points out that while Harry was off chasing Horcruxes, he'd been fairly effectively leading the resistance in his absence and Luna points out that she'd worked out how to find the penultimate Horcrux before Harry (or even Hermione) had.
In The Wheel of Time, Black Sister Elza Penfell destroys one of her superiors (though she is unaware that he is her superior because he is in disguise) who is trying to kill Rand al'Thor, because she believes Rand must stay alive long enough to face (and be destroyed by) the Dark One.
Demandred and Padan Fain also have this attitude about Rand. None of the three succeed; the Dark One is defeated and resealed by Rand, and Demandred and Fain get taken out by other people before having a chance to face him.
In one of the BattleTech novels, Phelan Kell gives a speech that typifies this trope: "I saved him because if Vlad is going to die, it will be at my hands."
Sandor Clegane in A Song of Ice and Fire gets particularly irritated when anyone apart from him expresses an interest in killing his brother, the pathologically nasty Gregor Clegane, who permanently disfigured Sandor in his youth. As Littlefinger explains to Ned, "Gregor was his to loathe, not yours to kill."
In Stardust, all the heirs of Stormhold are trying to kill each other and this is right and proper. But when an outsider kills one, the victim's ghost demands his remaining brother avenge him. The brother immediately sets out to do so. And gets killed in the process.
Inverted: In Charles Stross' The Jennifer Morgue a villain sets up a geas making him vulnerable to only one hero, one who suffers under the handicap of being (in the novel's universe) fictional.
In Dan Abnett's Gaunts Ghosts short story "In Remembrance", Rawne tells the unconscious Gaunt that he can't die, because Rawne wants to be the one to kill him.
Also, in Ghostmaker, Rawne and Gaunt team up to survive the blizzard and the Ork assault. Despite Rawne hating Gaunt's guts, he helps Gaunt and himself partly for this reason. (The other part was I Owe You My LifeWhether I Like It Or Not.)
In Greg Rucka's second Perfect Dark novel, the Big Bag does this in concern to Joanna Dark, the hero. Joanna points out that she herself is ill, mainly because of a barely-healed gunshot wound to the stomach. The Big Bad pulls a gun, reverses and blam. Now both have a gunshot wound to the stomach. Now nobody can say the fight was unequal.
Agrus Kos from the Ravnica Block Magic The Gathering novels is an unlucky police officer chosen by the Guild-master of the hidden guild Dimir to be the only one who can defeat him, by making him the only one who can defeat him, which would nullify the guildpact, essentially making him a Quantum Immortal... until Agrus finds YET ANOTHER LOOPHOLE and instead of killing him, merely arrests him.
In the epic Mahabharata, Karna after The Reveal vows to his mother that either he will kill his half-sibling and eternal rival, Arjuna or he would be killed by him. For Karna, Arjuna is the only one allowed to defeat him.
Happens all the time between Canim and Alerans in Codex Alera. In Canim culture, a respected enemy is considered more valuable than a friend. Many times in the fourth and fifth books, if certain situations were taking place entirely between humans or even if the roles of humans and Canim were reversed, characters would justify their actions to rivals or authority figures by calling an Aleran a friend or ally who needs their help. Since they're Canim or talking to Canim, though, they go to great lengths to make it clear that they don't like helping the Aleran, but neither another rival nor the Big Bad can be allowed to kill them, so...
Another interesting aspect of the concept of gadara (sort of a combination of this trope, Worthy Opponent and Friendly Enemy) is that once you have declared that you are the only one who is allowed to defeat someone, you have to defend your claim on your gadara's life from any other, less friendly enemies your gadara might have.
Redwall's Marlfoxes are a Big Screwed-Up Family with no problems killing each other off at the drop of a hat. However, they have a strict "blood for blood" rule if an outsider kills one of them, and they will enthusiastically enforce this rule.
When the Phoners in Stephen King's Cell tell you not to touch one of their enemies, they mean that if you kill one of them, they'll make an example of you.
Exploited in Pact, during Blake Thorburn's fight with Conquest, when Blake is near-fatally injured at the hands of a third party. Knowing that what Conquest wants more than anything is the victory of truly defeating him, and that killing him at this point would be more of a Mercy Kill instead of a victory, Blake successfully bargains for three days for himself and his allies to rest and recover, at which point Conquest can say that killing Blake was his doing.
The Master from Doctor Who, despite often trying to kill the Doctor, agrees to try and save his life in "The Five Doctors", because (in his words) "the cosmos without the Doctor scarcely bears thinking about."
This is actually a recurring trait with them. During Trial of the Time Lord, the Master actually flat out says he'll help defeat the Valeyard because he wanted to defeat the Doctor himself. Even in the End of Time, the Master, sacrifices himself to save the Doctor from the Time Lords. Add in the epic amounts of Foe Yay between these two, and it might just be that all the Master wants is to "keep" the Doctor all for himself, which he actually does in Sound of the Drums/Last of the Time Lords!
This causes the title character in Dexter to save the Trinity Killer's life which comes back to bite him hard in the end.
Inversion: Lindsey seemed to think that Angel had to be the one to kill him, as his last words, after being shot by Lorne, were "You kill me? A flunky?! I'm not just...Angel...kills me. You don't... Angel..."
This is a common Joss Whedon technique: a character may think that only one person is able to defeat them, but Joss delights in pointing out that unless they're supernaturally powerful (and sometimes not even then), they die just the same from one gun as another.
The reason Connor defended Angel from Linwood's commandos in "Tomorrow".
In Kamen Rider Black villain Birugenia decides that he will be the one to defeat Kamen Rider and constantly gets in the way of his allies plans when it seems possible they might actually defeat Kamen Rider.
Damon of the series The Vampire Diaries has this attitude toward Stefan in the show. He saved Stefan's life because he didn't want anyone else to have the pleasure of killing him.
Given the arc of their relationship over the course of the series, it seems this was really an excuse; he actually didn't want his little brother dead, but was keeping up the "I hate you so much" schtick.
The intellectually vain Detective McNulty of The Wire has this attitude towards Stringer Bell, his Worthy Opponent on the other side of the law, to the point where he's all but brokenhearted when Stringer is killed by rival gangsters before he can bring him down.
A version of this shows up in the Babylon 5 episode "The Coming Of Shadows", in which G'Kar was about to assassinate the Centauri Emperor at a reception, but was interrupted when the Emperor keeled over from illness. He later complains about this to his contact back home, and hopes that the Emperor will recover so that he'll have an opportunity to try again later.
Another version comes from Londo stating early on that, due a prophetic dream, he knows that he and G'Kar will strangle each other to death. In a moment of rage Londo tries and defy that by grabbing a gun to murder G'Kar, and later G'Kar tries and kill Londo by not saving both of their lives in a dangerous situation, but both times external factors make sure the prophecy will come to pass.
There's an episode of Everybody Loves Raymond where Ray ends up in a rivalry against Peggy, the obnoxious den mother of the Girl Scouts Expy that Ray's daughter is a member of (to elaborate, Peggy has a reputation for bullying the other scouts' parents in order to get choice spots to sell cookies, thus guaranteeing that she herself will win the top prizes). Ray is apparently the first parent to actually try to stand up to her, and she reacts by trying to pull his sales table away from him in public and push him onto the ground. Ray's wife Debra shows up and tells Peggy to back off. At first one is inclined to think that Debra, who usually bullies and abuses Ray a lot herself, is going to finally redeem herself...but then as soon as Peggy is out of sight, Debra goes back to bullying Ray. Apparently, she didn't really object to Ray being bullied (indeed, in later episodes such as "A Date for Peter", she uses the events of this episode to make fun of him in public)...she merely objected to someone else bullying him, because she apparently views him as being her property (emphasized when she tells Peggy "if you've got a problem, you bring it up with me" as if Ray isn't his own person, and is merely just one of Debra's possessions).
In Stargate Atlantis, Ronon wants to be the one to defeat a Wraith bruiser, telling Sheppard that he'd kill him if Sheppard killed the Wraith before Ronon. The Wraith beats the living crap out of Ronon until Rodney and Carson hit him with a missile. They're both profoundly apprehensive about it...until Ronon gives Carson a big hug and thanks him instead.
In Power Rangers in Space the Psycho Rangers had this attitude towards the real Rangers, which is no surprise, seeing as each one was programmed to defeat his or her counterpart. It was, in fact their greatest weakness in more ways than one; the hate that each of them showed towards their counterpart kept them from cooperating with each other at all (as opposed to the true Rangers, who were very good at doing so), and they were so obsessed with defeating the Rangers that when Astronama truly had them at their mercy by turning them all into computer chips, they ruined her plan by turning them back to normal simply so they could fight them. Most of them eventually failed, but Psycho Pink actually seemed to succeed in killing one of the Pink Rangers in Power Rangers Lost Galaxy; she turned up alive in the series finale, but it was still a victory.)
In Power Rangers Samurai Deker has this attitude towards the Red Ranger Jayden, considering him to be the ideal opponent in the Ultimate Duel therefore he even goes as far as reviving Jayden from poisoning to make sure he is fit enough for a fight between them. He even lampshades it to Jayden at one point.
Deker: Saving you has become an increasingly annoying habit. Though, it is one I intend to sub break.
In an interview with William Campbell, the actor who played the Klingon Koloth in the Star Trek: The Original Series episode "The Trouble with Tribbles", Campbell revealed that the original plan was to make Koloth a recurring villain, sort of the Klingons' opposite number to Kirk. He saw Koloth as someone who respected Kirk as an adversary, and would even protect Kirk from other assailants on occasion, explaining that "No one can kill you but me."
Lucifer has this view about Michael in Supernatural. Going so far as to blow up Castiel for throwing a Molotov Cocktail of holy fire.
This is Patrick Jane's attitude about Red John in The Mentalist, but not to the point of helping him out of other scrapes.
The feeling is mutual, and Red John is willing to extend his own efforts to protect Patrick from those less worthy.
A conversation between Beckett and Senator Bracken, the man who, years earlier, hired a hitman to kill her mother, upon being forced to defend the man against an unknown sniper:
Bracken A shooter... on the loose... me in the crosshairs... must be a dream come true for you.
Beckett In my dreams, I'm the one who gets to pull the trigger.
Zamusha, an Alien Swordsman in Ultraman Mebius came to Earth solely to fight Tsurugi (semi-formerly, Ultraman Hikari) despite having endangered the planet while fighting two rogues. After being beaten by Mebius and Tsurugi in battle, Zamusha vowes to carrying out his plan to kill Mebius and Hikari someday. Towards the series finale, Zamusha returns to save GUYS from a rogue Imperializer (and latter Alien Emperor,) from killing them simply because a weakened Mirai was amongst them.
The Doctor from Agents Of Shield is obsessed with getting his revenge on Whitehall for killing his wife. When Coulson kills him just as the Doctor is about to begin their fight, the Doctor has a Villainous Breakdown and shifts all of his anger to Coulson.
Also, Jr. and Albedo - this is stated explicitly in the second game
Bass is actually a playable character in a few Mega Man games because he doesn't want any of Wily's other robots to kill Mega Man before him (and because he sees their creation as an insult to him).
Inversion from the Suikoden series: Clive, Kelley, and Elza started as Two Guys and a Girl being raised by a group of assassins. The story ends pretty much like you'd expect: Elza, The Chick, on the run from Clive after being framed for Kelley's murder. What makes this different: she secretly wants to be caught, making this a case of "You're The Only One Allowed to Defeat Me". (Naming all the times the trope was played straight in the series could very well overtake the page. The biggest example being Pesmerga and Yuber, who have apparently been bitter enemies for centuries.)
Scorpion wants to kill Sub-Zero in Mortal Kombat, to pay him back for killing him (as illustrated in Mortal Kombat Mythologies Sub Zero). At the end of the first game, he succeeds. Then after discovering that Sub-Zero had a younger brother, Scorpion then subverted this to some extent by vowing to be the younger Sub-Zero's guardian as an act of penance for killing his brother. Quan-Chi then reverts it back to normal for Mortal Kombat 4 and MK vs. DC by convincing Scorpion that this Sub-Zero was responsible for the death of his clan and family. However, by the 5th game, Scorpion learns the truth and the two end up having another truce.
In Star Fox, The Rival Wolf acts this way towards Fox. The games thus far have never really provided a reason for this though, and the more recent games used it more as an excuse for Enemy Mine situations (involving Wolf saving Fox so that he can "tan his hide" after the big threat has been dealt with). They still clash in every game for no real reason other than this, of course. Maybe he's just a sore loser.
Star Fox Command explains this a little better. Because they flew under Andross's colors in other games, Wolf and his teammates have had a bounty placed on their heads by Corneria. And since Starfox used to do mercenary work for the Cornerian army, Wolf can't really be blamed for being a little bit defensive whenever Fox and his team shows up, even if they 'should' be working together against the Big Bad. This doesn't make it any less corny when the big Enemy Mine moment actually happens.
Heather in Silent Hill 3, when Claudia eats the baby God fetus, and proceeds to give birth to it (seriously, how does that work?). Shortly afterwards, the God presumably killed Claudia, as she was pulled down the hole at over 300 mph. Heather then proceeds to state, "No! You can't die! I wanted to kill you!"
A variant of this is used in two of the Mario games. In Super Mario RPG, if Mario tries to leave Marrymore through the alternate exit after saving Peach and says no to everyone, Bowser will eventually come out and say "No one, NO ONE, is authorized to kidnap the Princess except ME! It's just wouldn't be right!" In Paper Mario: The Thousand-Year Door, upon finding out that Peach was kidnapped by the X-Nauts, he says practically the same thing, going as far to say it's a rule against it.
In Fortune Street, he angrily tells a victorious player that Mario's the only one who's supposed to beat him. He then backpedals and says that no one's supposed to beat him at all.
The ending of Mario Super Sluggers shows Wario and Waluigi putting a Bullet Bill in a fireworks cannon aimed at Mario. Bowser then leaps in and hits the Bullet Bill with his baseball bat saving Mario in the process.
In Metal Gear Solid, Liquid was the only one allowed to kill his father, but Snake had already killed the old man twice. This makes Liquid seriously pissed, and he takes up his father's mantle of destroying the world to prove himself better than his father once and for all. And to kill Snake in revenge. Those two goals go hand in hand.
Ocelot and Naked Snake in Metal Gear Solid 3: Snake Eater. Especially noticeable when Snake chose to jump off the waterfall rather than get shot by Ocelot. A Slow Motion Fall scene occurred, while Ocelot ran in slow motion towards Snake while calling his name. Right after, Ocelot whispered, "Don't die on me yet." Subverted:The Stinger reveals that Ocelot was Snake's contact ADAM all along, and most of the times he spared or arranged for Snake's life to be spared, he was fufilling his mission.
Yeah. He got over that fast. Especially since he ends up joining Big Boss after Major Zero brings them all together and in Metal Gear Solid 4: Guns of the Patriots Big Momma says that Ocelot was fighting for Big Boss. Kind of a Heel-Face Turn right there. From wanting to kill him to following him around.
Also inverted in the case of Vamp in Metal Gear Solid 4: Guns of the Patriots: After Raiden and Vamp's duel in Act 2, to which Vamp was impaled by Raiden, Vamp asks with ecstasy "Could you be the one to finally finish me?"
In the first Samurai Warriors, the wife of Oda Nobunaga cites this as the reason for being at his side (historically, there had been suspicions that she was married to Nobunaga so she'd assassinate him), but in both of her endings she seems to have more or less given up on the idea of actually killing him (except to threaten it every once in a while halfheartedly as a very kinky form of foreplay). In later games she's pretty much head over heels in love with him from the get go, though still very much a sadist.
You'd better stay alive until the day I'll kill you.
In Super Robot Wars Original GenerationZengar's Grungust Type-3 has just been pretty well demolished by one of the Inspectors, when who should show up but Wodan Ymir, Zengar's doppelganger from a different dimension. Everyone thinks he's here to finish Zengar off...and instead he attacks the Inspector. A few minutes later, when Zengar gets into the Daizengar, only to find none of its weapons work, Wodan once again makes the save by throwing him the Type-3 Zankantou. His whole reason for doing this, as he points out, is that he will be the one to defeat Zengar, so that he can stop being one of the Shadow Mirrors' soulless puppets and finally be his true self.
Frog to Magus in Chrono Trigger, if he's in your party when they try to stop Lavos and Magus goes down. Later on if you chose to fight Magus, he'll insist on doing it alone.
In Yakuza, one of Goro Majima's henchmen attempts to kill Kazuma Kiryuu with a dagger while his back is turned. Majima throws himself in front of the blade, taking a near-fatal stab wound to the gut rather than risk Kiryuu dying at anyone's hands but his own.
Borderlands has the standard bandit mooks yell, "Nobody shoots my buddies but me!".
Also, in the mission where you have to rescue Lucky:
Scooter: "Lucky's a friend, and by friend I mean asshole who broke all my momma's ladyparts. Could you go rescue him so I can kill him on a later occasion?"
In the General Knoxx DLC, we find out that this wasn't an empty threat. "Don't act so surprised, I told you I was gonna do it!"
Borderlands 2 has Handsome Jack trying to kill the Vault Hunters by posting a huge bounty on them for most of the game. However, at one point this bounty is rescinded because you helped his daughter, Angel, kill herself - and Jack states that he wants to be the one to kill them. No one else.
Later on, Handsome Jack posts a bounty on the bounty board asking the Vault Hunters to kill themselves for a king's ransom. If you do, he'll revive them and give the promised reward, but wants you to know that you are a sell-out.
In Mega Man Zero, this pretty much sums up Leviathan and Fafnir's mindset towards Zero.
Final Fantasy VII has Cloud and Sephiroth. In fact, you have an anticlimactic battle where Sephiroth just stands there, waiting for Cloud to use Omnislash (waiting long enough will have Sephiroth attack Cloud with a relatively weak attack, to which Cloud will then finish him off, however).
It's also done literally, as well: Sephiroth has specifically stated Cloud is the "only one" who can destroy him. In every appearance he has had, he has been fought by Sora, The Warrior of Light, Firion, Tidus, Cecil (all three offscreen), Tifa, Zack Fair, and countless others...out of all of them, only Sora and the Warrior beat him in a sword duel when Sephiroth went all out, but with the cutscenes that follow after Sephiroth makes it clear that they can't truly beat him.
BlazBlue has Jin Kisaragi who constantly says that he is the only one who must defeat his Older Brother and Rival, Ragna The Bloodedge.
Kokonoe is an even worse version. She had to be the one to frag Terumi, no one else. Even if she had to use nukes.
The 8-Bit version of Sonic the Hedgehog 2, sees Dr. Robotnik catch Sonic the instant before he falls into a lava pit only to drop him into an adjacent, much less deadly self-devised trap. Can be seen at 1:50 here.
Shadow the Hedgehog has also pulled this off. He expresses regret whenever it seems Sonic is defeated by somebody other than him in various games.
Metal Sonic, who is programmed to kill Sonic by Dr. Eggman anyways, takes this even further in Sonic Heroes when he turns into Metal Madness/Overlord, extremely obsessed in not only defeating Sonic but also showing that he's the real Sonic.
Applies to Fayt and Albel of Star Ocean Till The End Of Time as Albel saves Fayt's ass on one occasion because "You worms…are…my prey… I’m not going to let…anyone else have you…".
Lili towards Asuka in Tekken. She even jumps on a certain android attacking Asuka and screams "I'm the only person allowed to defeat you!" Keep in mind this android has chainsaws for arms, and previously cut a limo in half while chasing them; lengthways. Therefore could have easily killed Lili. Any wonder why this is a budding Crack Pairing?
Hwoarang is this to Jin Kazama as well as Jin is the only person who defeat him (Jin himself knows why Hwoarang wants to be that guy but he considers any rivalry there to be small-time compared to his conflict with his Big Screwed-Up Family).
Bastion, the hero of Vanguard Bandits is the only one who can land the killing blow on the Final Boss. If anyone else does it, he comes back with half his HP restored.
Aveline says this regarding Isabela in Dragon Age II at the end of Act II, when the Arishok states that they'll only leave Kirkwall with both Isabela and the Tome she stole from them.
Most of the Arishok's dialogue implies that he considers Hawke the only person in Kirkwall worthy of taking him down, especially if you've earned his respect. One interpretation of his motives for invading Kirkwall and offering a Duel to the Death against Hawke was a way of performing Suicide by Cop, freeing him from his own principles and demand of the Qun that had bound him in Kirkwall for four long years.
Raz in Psychonauts tells his camp councilors this about the Big Bad right before they send him away from the battle, claiming that "the grownups should handle this." He's understandably pissed off later.
In Ratchet & Clank: All 4 One when you play as Qwark and Dr. Nefarious, if Qwark is killed, Nefarious will yell, "No one kills Qwark but me!" Also, Nefarious saves Qwark from being possessed by the Big Bad towards the end of the game.
According to supplementary materials, Auroran warriors in EV Nova sometimes acquire "blood enemies" from one of the rival Houses. The blood enemy will seek to ensure their rival's survival until they get an opportunity to face them in fair, open combat.
In Shin Super Robot Wars, Ghostelo wants to be the only one to beat Eiji Asuka. The Daikyu Maryuu team takes advantage of this during one scenario, Ghostelo appears and smacks the supercharged Zakarl out of the way because the Layzner is his prey. Ghostelo has made himself an enemy out of his own allies, and the SRX Team realize they've only got one chance to defeat Lu Cain, something that hasn't succeeded until now, but likely won't succeed yet, causing Daimoji to sortie the rest of the crew, including the Shin Getter Robo.
In Uncharted: Drake's Fortune, Nathan Drake and Eddie may feel this way. This is shown in the chapter where Drake and Eddie team up to fight The Descendants, although this might've just been the fact that they teamed up to avoid dying themselves.
Eddie: "If we don't make it out of this, I just want you to know... I hate your guts."
Nate: "Yeah, Likewise pal, NOW LET'S DO THIS!"
Mass Effect 2: If you successfully break up the argument between Jack and Miranda, Jack will say she'll keep Miranda alive during the fight with the Collectors so she can have the pleasure of killing Miranda herself. Thankfully, they never get around to it, and eventually evolve into a Vitriolic Best Buds relationship.
Fate/stay night has an odd version, where Gilgamesh seems to believe that he's the only one allowed to defeat anybody. He considers the world and everything in it his property, which means that he is allowed to do anything he wants to anybody he wants, but if someone else starts killing people en masse he becomes infuriated and will hunt them down for attacking his 'subjects'.
In The Order of the Stick, Belkar helps rescuing Elan from bandits who've captured him, because "if anyone is going to get XP from him, it'll be me".
Also lampshaded when Crystal is more than willing to let Haley go so that she can take another level of Assassin from all the free XP she gets in order to be at exactly the same level as Haley whenever they meet.
Elan's father Tarquin inverts this. Elan is the only one he will allow to defeat him — a fair chance anyway — since that would make an epic story. Tarquin is pretty Dangerously Genre Savvy otherwise and takes precautions against letting anyone else get close enough to even try to kill him — especially not Nale The Unfavorite.
Speaking of Nale, his girlfriend Sabine won't let anyone except her kill Haley, and gets extremely mad at him when he (posing as Elan), seduces Haley with the intent to kill her. Notably, being a succubus, Sabine is more upset about the attempted murder than the seduction, and is only placated when Nale ensures her that he was only going to capture Haley so they could "romantically" kill her together.
Malack insists that Durkon be left for to "handle" when it looks like their parties will be fighting each other. It's not clear whether handling him means removing him from the battlefield, killing him, or turning him into a vampire like himself, which is what he ends up doing.
Narbonic: When Helen hears that the Dave Conspiracy has hired Mell to kill Dr. Narbon, she exclaims, "I don't care if they are a powerful top-secret conspiracy! No one takes out a hit on my mother! Her head is mine, darn it!"
Parodied in Looking for Group when Benny heals the mortally wounded man who killed her lover, just long enough to smash his head with a mace.
In Brawl in the Family, Meta-Knight has Kirby cornered when Dedede attacks him from behind, saying "This is MY battle." To his dismay, Kirby hugs him in gratitude.
Last Res0rt has Jason Spades filling out this role to a tee:
On Atop the Fourth Wall Mecha-Kara attacks Cable for this reason, as he was out to kill Linkara after he teleported him away and stole his gun.
The Joker Blogs - The title character, unsurprisingly. While his priority in this series is stalking Doctor Quinzel, he doesn't appreciate another character taking a shot at Batman. "You do NOT get to do that!"
Kim Possible: Shego's only mildly annoyed (by her standards, anyway) when she gets imprisoned by Drakken's new alien sidekick. But when she realizes that the new girl has a chance of beating Kim, that pisses her off enough to break free and put a stop to it.
Shego: "I am not gonna let this she-thing just waltz in and destroy Kimmie! That'smyjob! Ya hear me?"
Danny Phantom. Skulker is constantly agreeing to work together with Danny and co. in order to save the world, Ghost Zone, or anything else on a massive enough scale to matter. Normally these occurrences are justified by the fact that EVERYONE would die if all the bad guys didn't help, but he tries to make excuses anyway.
Transformers. Especially in the Unicron Trilogy, it seems like Megatron always seems to single out Optimus in combat. Also, in the third series of The Original Series, Cyclonus occasionally saved the life of Ultra Magnus because so he could be the one to kill him.
The Megatron of Transformers Prime has the same complex, and does not react well to Starscream trying to deal with the Autobots behind his back. "NO ONE RIDS ME OF OPTIMUS PRIME BUT ME! DO YOU UNDERSTAND?! DO YOU!"
Although, Prime Megatron seems more reasonable about this than most - as long as he ordered Optimus to be killed, it's fine.
Forget the Optimus/Megatron rivalry...Starscream, a Dragon so treacherous he's got his own trope, is so obsessed with being the only one allowed to defeat, defy or otherwise annoy Megatron; he goes into an apparently righteous rage when Skyfire, Thrust, Sideways and the Constructicons even allude to an Enemy Civil War. Hell, he even saves Megatron from someone else's betrayal in "Atlantis, Arise!" It's a safe bet that he only went along with Blitzwing and Astrotrain's plan in "Triple Takeover" because they let him be the one to lead Megatron into the trap.
Motormaster also feels this way about Optimus Prime, as he feels that Prime is The Rival to his self-professed title of "King of the Road".
The Venture Bros.: The Monarch and Baron Ünderbheit compete for this role against Thaddeus Venture, with The Monarch's unexplained obsession forming the basis of a comical deconstruction of this trope as a form of addiction.
In general, the Guild of Calamitous Intent arranges these through the villains and protagonists so that there is supposed to be only one person you can defeat you. This is what gets the Monarch in trouble with the Guild originally, he wasn't the person who was allowed to defeat Venture.
Proto Man and Mega Man in the US series of Mega Man.
Specifically, in the very first episode when Wily wants to blast Mega Man, Proto Man smashes the necessary button before Wily can press it and tells Wily that Mega is "his". And in "Future Shock", Mega's attempt to use the stolen time machine (long story) is thwarted by a low-powered bomb planted in the cockpit—by Proto Man, no less; it was only so he and Mega could have a proper fight. And don't get me started on "Bro Bots"...
Prince Zuko in Avatar: The Last Airbender needs to capture Aang to regain his honor and be allowed back into the Fire Nation. This has led him to rescue Aang from other Fire Nation soldiers.
Also somewhat inverted later when Zuko decides the one to kill his father has to be Aang, not himself, even when he had the opportunity.
Likewise, in the series finale, Iroh refuses to fight Fire Lord Ozai (despite being the only other person powerful enough to beat him), because the Avatar has to do it himself to truly achieve peace. The Avatar carries considerably more clout than two exiles, who would be treated as such.
In a literal sense, Demona and Macbeth in Gargoyles, who due to sorcery are immortal unless one kills the other.
Demona, also feels this way about Goliath for the most part. There is even an episode, where Demona attacks Elisa Maza to draw Goliath out in the open and hunts him for the rest of the episode. If not for Hudson being with him, Demona would have succeeded. This would become enhanced once Demona realized she had a daughter named Angela.
In more than one episode, the Joker displays this attitude towards Batman: only he gets to kill Batman, or, failing that, he (the Joker) dies in one last climactic battle between the two. He even tried to kill a lowly henchman who was thought to have killed the Bat.
Harley: That's a real gasser, huh, Mistah J? Joker: (grabs her) I give the punchlines around here, got it? Harley: (scared) Yes sir.
In the comic book Mad Love, which inspired the episode, the Joker throws poor Harley out of a fifth-story window after arriving on the scene - in the episode, it's "only" the third-story.
There is also "The Man Who Killed Batman", where Joker becomes depressed after learning of Batman's death. He stages a jewelry store robbery with the man credited for his death, hoping Batman will show up. When he doesn't, he clearly shows anger towards the man credited for his death and has him thrown into a pit of acid. Batman, however, is still alive and saves him at the last minute without making Joker aware.
This is carried over to The Batman, where Joker actually goes so far as to incapacitate Evil CounterpartEvil DuoWrath and Scorn with Joker Gas (after the duo had gone out of their way to repeatedly help the villains of Gotham in escaping Batman and Robin, Joker included) because they threatened to reveal Batman's secret identity to the entirety of Arkham.
As Batman Beyond would demonstrate, Joker would die twice to someone other than what he would consider Batman, and he wouldn't get the real Batman either.
Also displayed in Batman: The Brave and the Bold, during an episode where Batman is fighting his evil alternate self Owlman and has teamed up with the Joker to give himself an advantage. At one point, the Joker rescues Bats from being killed.
In the Legion Of Superheroes, Superman clone Superman X was literally created to fight galactic despot Imperiex; when Brainiac 5 wiped him out with the wave of a hand, X seemed a lot more disturbed by the appropriation of his nemesis than by Brainy's Face-Heel Turn or his subsequent "murder" of Superman.
Batman Beyond: this was the reason given by the Stalker after he rescued Terry.
In one episode of Beavis And Butthead, a candy salesman gets mad at the duo for eating all the product they were supposed to sell. When he begins attacking the two of them for it, the teacher, Coach Buzzcut, steps in and beats him up, proclaiming, "This is MY class, I do the ass-kicking around here!"
Zim of Invader Zim is out to destroy the Earth, but won't stop at anything to thwart anyone else who tries to do it competently.
Dib is also obsessed with Zim.
In the Family Guy episode "And Then There Were Fewer", Diane Simmons attempts to kill Lois. However, Stewie shoots Diane so that he can be the one to kill Lois.
Angelica: "Listen, lady, no one messes with my dumb babies 'cept me!"
A less Foe Yay and more innocent version of this occurs in Arthur. In one episode, the moose kid, George, is being picked on by bullies. Then Binky, the established bully of the series approaches them and says, "Hey! You can't pick on that kid! He's in my class!" George looks relieved, until Binky adds, "Only I get to pick on him!"
In the South Park episode "Pre-School" Trent Boyett, who Cartman describes as the "Meanest, dirtiest, toughest kid in the world, super-pissed off at US Trent Boyett", is released from Juvenile Hall. The boys framed him for an accident and he is out for revenge. As they can't go to their parents as they lied years ago and he beat up the sixth graders, Stan decides to go to Shelly for help.
Shelly: Calm down, turd! No Juvenile Hall turd is going to kill you. That's my job.
Cartman's obsession with Kyle that easily clear Foe Yay.
In Aladdin: The Series, a genie hunter hired by Mozenrath leads Mozenrath to believe that Aladdin was killed in the process of hunting the genie, because he's turned against the sorcerer and wants Aladdin to be able to come and rescue Genie. Mozenrath hates this, to the point where when Aladdin shows up, he says he's glad Aladdin is alive so he can kill him.
A non-violent variation in Sofia the First: when Miss Nettle returns in disguise with a scheme to steal Sofia's amulet, our heroine wins the aid of Cedric when she tells him Miss Nettle's after the amulet - which, of course, he himself has been after since day one:
Cedric: No one can have that amulet but me!...er, I mean you, Princess!