The protagonists of any given story are, of course, larger than life. Expecting the various extras and Red Shirts to actually accomplish something noteworthy would probably come off as anticlimactic. But even among the various named characters, there's clearly a hierarchy involved. That hierarchy, among other things, mandates that only the protagonist can achieve certain things. Nowhere is this more clear than when battling the Big Bad, often Because Destiny Says So.
Anyone other than the main character will almost inevitably fail to defeat the Big Bad. It's a rule of drama. Any poor schmuck who tries, be he The Lancer, the hero's closest friend, the hero's Love Interest, or a random soldier, will almost certainly be cut down without the Big Bad breaking much of a sweat. There will be gloating. There may be slow-motion footage of their fall, and dramatic music.
Camera pans to the hero. Cue the Unstoppable Rage.
The reason writers came up with the Hero Secret Service, which gives the rest of the Five-Man Band something important to do without directly taking part in The Hero's mission.
Often overlaps with the Chosen One, as well as This Is Something He's Got to Do Himself. Differs from The Only One Allowed to Defeat You in that it's a matter of raw ability, not personal preference.
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Anime and Manga
It's almost an iron-clad rule that in Shōnen anime, the Big Bad can only be defeated by the plucky young main hero. The hero's friends are usually capable of taking care of the Big Bad's Evil Minions, but if they try to take out the Big Bad by themselves, they'll usually fail. (Oddly enough, it seems the more powerful and experienced the hero's friend is, the more spectacularly they will fall in their fight against the Big Bad.)
Repeatedly Lampshaded in the third season of Yu-Gi-Oh! GX. Whenever Evil Minions like the Masked Knights and other zombies challenge Judai, Johan and the others always step in and remind him that, as The Hero, he needs to wait and save his strength for his inevitable battle with the Big Bad.
Almost subverted in the original series: Jounouchi (The Lancer) gets into a position where Marik has no cards at all to protect himself, a single direct attack will win it, and he has a monster strong enough to do it. However, before he can declare his attack and win Jounouchi collapses from exhaustion, and Marik wins on a technicality when Jounouchi is declared unable to continue the duel. That he was seconds way from defeating Marik was a shock to every member of the cast, including Marik himself who began to freak out when he realized he was going to lose and admits Jounouchi put up a lot better of a fight than he expected. (In fact, this was the one time that Kaiba said anything about Jonouchi that even came close to being a compliment.)
Subverted in Rave Master with the Big Bad Hardner. The Hero Haru Glory was able to defeat this foe only because Hardner had already fought and defeated one of Haru's friends in battle immediately before Haru arrived. The two fights back-to-back wore down the regeneration abilities Hardner possessed which rendered him virtually immortal, leaving him vulnerable to take more damage than he could heal.
Justified in Shootfighter Tekken by the main hero's father having become a pacifist at this point, but since his son shows an unstoppable drive to face down the Big Bad anyway, he tutors him specifically to do so. Lampshaded by other characters fighting other mooks, or being ambushed by them, but everyone in the entire series knowing and stating only the hero can take on Iron Kiba. In hospital scenes and conversations between trainers, it's acknowledged even further. Only Kiba does not seem to know, as he is completely focused on the hero's father instead because of their battle in the past, and views the kid as being little more than a training exercise. Interestingly, he was nearly proven right, as the hero didn't take the fight seriously at first, and the first 60 seconds was essentially a cock-measuring contest of insults and posturing, each believing the other wasn't a serious challenge.
Happens all the time in Bleach, but most strangely in the Captain Amagai Filler Arc. Ichigo has almost nothing to do with Amagai; he only meets him for the first time shortly before he fights and defeats him. And this is in a city filled with Bad Ass captains. The only possible reason he would have for being the one to kill him is because he can overcome Amagai's zanpakuto-suppressing ability.
In Inuyasha, Big Bad Naraku is threatening the whole region, yet there are only three fighters outside of the protagonist's foursome that fight the big bad for more than a single scene of them getting slaughtered. Slightly subverted towards the end, where a Buddhist priest manages to take a pot-shot at one of the greater villains, and a Shinto monk actually defeats another (who ends up taking over his body). A pity because the villain is Naraku's heart, and if the guy had killed him instead of absorbing him, the series Big Bad would have ended.
Played with in Monster. Dr. Tenma and Nina are both trying to find and kill the eponymous Monster Johan: Tenma because he saved Johan's life, Nina because she was the one who shot him the first time, and both because neither wants the other to have blood on their hands. In the end, it's a vengeful father who shoots Johan...and Tenma is the one who saves Johan's life once again, thus inverting the trope.
Vocally invoked by Lina Inverse in the Slayers OVA Jeffrey's Knighthood. This is so that Jeffrey's mother will not beat them up when they leave Jeffrey behind. They're not ditching The Load, they're allowing him to prepare while they carve a path to the Big Bad that only he can best.
A variation of this occurs in Cowboy Bebop between Spike Spiegel and Vicious, as each declares that only he can kill the other. In the final episode Vicious dies for sure, but whether Spike survives or not is left uncertain.
A very rare inversion in With Strings Attached. On the Plains of Death, only the secondary character The Hunter can destroy the Heart of Evil by stabbing it with his BFS. Paul merely clears the way for him.
Or so the Hunter says. Since Paul doesn't even try to hit the thing, who knows?
Justified in that the Hunter is the Hero in this world, and the four (otherwise the protagonists of the book) are just being escorted around.
In Shining Armor's side story of the Pony POV Series, Reznov eventually implies that Shining may be the only one who can defeat Makarov, due to Shining being a temporal anomaly, and thus immune to the abilities that would prevent anyone from fighting him.
One example occurred in the movie King Arthur, where Tristan's attempt to slay the Big Bad ended in his inevitable death.
James Bond usually uses this trope, with Bond being the person who kills the Big Bad. However, it's subverted in From Russia with Love (Rosa Klebb is shot by Tatiana), For Your Eyes Only (Kristatos is offed by Colombo and Gogol survives) and partly in Casino Royale (Le Chiffre is shot by an assassin, although his employer gets shot by Bond at the end. It's even more true in the novel).
Subverted once again in the original Teenage Mutant Ninja Turtles movie, in which the main protagonists, the four turtles, attempt to defeat the Big Bad Shredder, and one by one they fail utterly. Shredder can only be defeated by the intervention of the previously hermitic Obi Wan of the turtles, Splinter.
Subverted in the movie Throne of Blood. Washizu is killed by his own nameless Evil Minions - indeed we do not even see the faces of the archers who loose the first arrow and the last. In the original Macbeth the villain died in a duel with a major character; in Throne of Blood that character is completely absent.
In the second Transformers movie The Fallen can only be defeated by a Prime for ill-defined reasons.
Literary example: The Keep by F. Paul Wilson. A great undead villain reawakens from his sleep, and the only person who can stop him is on the other side of the world — fortunately, he's psychically attuned to the villain and promptly gets moving.
By Word of God subverted in that anyone with the equivalent of a shotgun could have killed Voldemort once his horcruxes were gone.
More or less subverted by the fact that it's Voldemort's Bond Villain Stupidity which made him allow Harry to defeat him, precisely because the prophecy said so (defeat meaning not only the final fight, but the entire quest).
Subverted in The Bourne Ultimatum (novel version) where, after David Webb/Jason Bourne spends the entire novel saying that only he'll be able to kill Carlos the Jackal, Carlos ends up drowning in a tunnel flooded by the Soviet agent who's working with Bourne.
In Esther Friesner's novel The Sherwood Game, a programmer creates a VR Robin Hood game, and creates a specific rule that his character is the only one who can kill the Sheriff of Nottingham. He comes to regret this when he has to play the game with the safeties off.
Legend of the Seeker spends quite a lot of time saying how Richard is destined to kill Darken Rahl with the Sword of Truth. It's the same in the book, except there he's explicitly told that the magic of Orden means he can't use the sword to do it.
Subverted in the Left Behind books as The Word of God demands that nobody can even defeat Satan and the Antichrist except for Jesus Christ. The Antichrist does get killed partway through the Tribulation, Because Destiny Says So, but as he is resurrected by the indwelling of Satan for the remainder of the Tribulation, the Christians during that time will have to wait for Jesus to come again in order for the Antichrist to be sent to the Lake of Fire.
Live Action TV
Averted in the finale of Buffy the Vampire Slayer, season 6, where Xander — often the comic relief of the Scoobies — talks a grieving, psychotic, high-on-magic Willow out of destroying the world. Buffy, meanwhile, is fighting an endless and pointless battle against Evil Minions.
Also averted by Season 5. Buffy might actually do the beating (with a shitton of help) but Giles does the killing.
Played straight in Angel - Connor is destined to kill Sahjhan, which appears to mean no one else can. When Sahjhan is trapped in a magical urn, his enemy Cyvus Vail insists that Connor be brought back to finish him off, anticipating that such things never hold the bad guy forever.
To be fair, Angel had tried to kill Sahjhan when he was corporeal, and the demon handed Angel his ass.
Particularly notable in 24, wherein any tac team which does not include Jack Bauer is certain to let the terrorist escape, shoot the wrong guy, be vaporized in a nuclear detonation, etc.
Subverted by Firefly, in "War Stories". Mal is battling The Dragon (well... a dragon, at any rate) when the Cavalry (Zoe, Jayne, and Wash) shows up. Jayne takes aim to shoot Niska's henchman, and Zoe stops him, saying "this is something the Captain's got to do for himself." Mal yells a panicked "no, it's not!" and the three rescuers take out the henchman all at once.
Played straight in The MovieSerenity, where Mal is the only person who can take on the Operative and stand a chance. It helps that he had a nerve cluster moved during the war, making the Operative's paralyzing move useless. And Mal is the only one who can piss him off.
Subverted to a degree in LOST. It seems a lot like Ben was the only one who could kill Jacob. Several others tried, and the Man in Black said that some kind of "loophole" was necessary.
In the play Macbeth, the title character is informed that "none of woman born/shall harm Macbeth". This makes him believe he was invincible. However, Macduff was born due to a c-section, making him the only one who could kill Macbeth.
A literal example in in the final chapter of Fire Emblem: Radiant Dawn: Only Ike is allowed to deal Ashera the killing blow. Otherwise she regenerates to full health.
Another literal example shows up in Fire Emblem Awakening, albeit with a twist. The only power capable of destroying Grima is his own. As your Avatar is Grima's vessel, this means only the Avatar can kill Grima.
In Starcraft, only Zeratul is able to kill the Cerebrates in the original game's penultimate mission, if the kill is done by any other unit the Cerebrate regenerates. Due to a glitch with location triggers it's possible for another unit to kill the Cerebrate for good while Zeratul is directly nearby, but the intent is for him to do it. In the expansion the player repeats the process in some missions using generic Dark Templar in place of Zeratul to the same effect.
Final Fantasy IV: The After Years, the player must confront the Dark Knight with Cecil and Golbez in the party, otherwise he instantly kills everyone. Once the required cinematic between the three occurs though, any party member can kill the boss.
Raziel and Kain in the Legacy of Kain series—Kain doesn't want to kill Raziel and as he learns more about Nosgoth Raziel loses his desire to kill Kain. However, the two are functionally immortal, so they're the only ones that can kill the other by virtue of Kain possessing the material version of the sword Soul Reaver that can imprison Raziel within it, and Raziel possessing the spectral version of the same sword that can bypass Kain's vampiric body and damage his spirit directly.
Every boss in Kingdom Hearts II: you can let your AI allies attack as much as you like when the boss is down to 1 HP, but until Sora personally lands a Finishing Move they won't die.
One boss in the first chapter of Xenosaga can only be killed if the final blow is landed by Shion or Junior. This is because it is connected to various mental issues of theirs stemming from the Miltia Incident.
In Tsukihime, the strange way in which Nrvnqsr's body is made up means it is nearly impossible to kill him: you have to kill all 666 of his familiars at once or he can regenerate them instantly. The protagonist, Shiki Tohno, on the other hand, has the explicit ability to kill things Deader Than Dead, making him uniquely suited to killing Nrvnqsr. If Shiki kills something, it stays dead. Period.
In the Sluggy Freelance arc "Dangerous Days," it's ultimately Torg, a pretty ordinary guy, who defeats Aylee, not Badass Longcoat Riff, not super-assassin Oasis, not Killer Rabbit Bun-Bun, not even the real Aylee, because Torg was the one with the most emotional involvement.
Also done in the "Oceans Unmoving" arc, where the only one who even stands a chance against Blacksoul is Bun-Bun, and the only one who stands a chance against Bun-Bun is Blacksoul. Makes sense when it's later revealed that Blacksoul is actually Bun-Bun from the future.
Lampshaded by Belkar of the Order of the Stick, who declines to finish off Crystal because he recognizes her as Haley's nemesis.
In the last few episodes of Red vs. Blue: Reconstruction, Washington makes it clear that only Church is capable of stopping the Meta. However, this isn't due to Church's fighting ability (which has never really been particularly good), but rather because Church is actually the Alpha A.I., and thus the only being capable of neutralizing the A.I.s that the Meta has merged with.
Storm Shadow in G.I. Joe: Renegades needs to be the one to kill Snake Eyes in order to avenge the death of his uncle, the Hard Master and save face with his clan. He's also the only member of Cobra with a chance of actually defeating him one-on-one.
Samurai Jack is a literal case. He is the only one who can slay Aku, because his sword is the only known weapon with the power to do so.