In The Big Bang Theory, Sheldon has declared Wil Wheaton his arch enemy since childhood. In "The Russian Rocket Reaction," Sheldon finally forgives Wheaton but immediately replaces him with Brent Spiner.
However his rival Barry Kripke comes closer to be his true Arch enemy, as there both scientists of the same field, both highly intelligent, and both hold a strong rivalry, barring a few moments of them putting it aside.
Gus Fring very much becomes this to Walter White on Breaking Bad. Interestingly throughout their feud we see signs that Gus and Walt really aren't that different, and in fact Walt admires Gus in many ways.
Gus Fring also had Hector Salamanca and Don Eladio Vuente. On the orders of Don Eladio, Hector shot his best friend (and possible lover), Max Arciniega, in cold blood before forcing him to look in the eyes of his corpse.
Hank Schrader views Heisenberg as his.
For a while in Buffy the Vampire Slayer, Harmony liked to think of herself as Buffy's Arch Enemy, but Buffy never took her quite that seriously. Buffy even laughed herself sick when Xander told her that Harmony and her minions could be a threat to her in the second episode of season 5.
The evil trio in season 6 consider themselves the archnemesis of Buffy, but... yeah... she doesn't take them seriously either — which turns out to be a pretty big mistake. Their leader, Warren, manages to screw with Buffy — mentally and emotionally — more than any other villain, save Angelus. He later incurred Willow's wrath entirely by accident, but from there on, they're vicious enemies, particularly in Season 8 (though he still despises Buffy).
Faith is probably the closest thing to playing this trope straight. Even after Faith's Heel–Face Turn when the two are supposed to be on the same side during Season 7 and the Season 8 comic, they can't help fighting and sometimes trying to kill each other.
Up until his Heel–Face Turn, Spike was this to Buffy and the rest of the Scoobies.
Another way to look at it is that Buffy has a different arch-enemy each season (due to each season having its own Big Bad).
Joss Whedon plays with this trope again in Angel. Lindsey thought he would be Angel's arch enemy, having been Angel's enemy since the show's first season and having spent half his life trying to beat Angel. Which probably why he seems personally offended by the fact that it's not Angel who kills him, but Lorne.
Lindsey:You kill me? [collapses] You... a flunky? I'm not just- Angel kills me. You- Angel...[dies]
Despite Angel not taking him seriously, Lindsey still would basically would be Angel's archenemy overall. No other individual character gave him as much trouble as Lindsey did, other than Lilah.
Lilah is sort of an archenemy to Cordelia Chase. Both of them are basically both The Dragon and Alpha Bitch for the good and evil sides, respectively, and get into numerous personal conflicts with one another. The most notable of these being when Lilah was directly responsible for Cordelia's worst experience on either show, when the demon Lilah employed hijacked Cordelia's visions to become way more painful than they already were, causing her own mind to completely wreck her body and almost kill her.
Stephen Colbert declared Korean pop star Rain his Arch Enemy for constantly besting him in Time Magazine's Top 100 Influential People.
The rivalry intensified after Rain soundly defeated Colbert in a Dance Battle.
Second place for his Arch Enemy is probably Representative Eleanor Holmes Norton of DC. They always fight whenever she is on, but it is not mean spirited.
On Criminal Minds Frank Breitkopf was Jason Gideon's Arch Enemy (and before Frank, the bomber who killed Gideon's old team and caused him to have a nervous breakdown), The Boston Reaper was Aaron Hotchner's, and Ian Doyle was Emily Prentiss'. One could make a case for Billy Flynn being Derek Morgan's, and Tobias Hankel being Spencer Reid's.
Flynn and Tobias both lasted for one two-part storyline, though Tobias ended up casting a shadow over Reid for a few episodes. With Billy Flynn, Morgan just said It's Personal partly because he felt frustrated after the Boston Reaper case (the Reaper once knocked Morgan out and spared Morgan only because he didn't feel like killing him) and was taking out that frustration on Flynn (who, like the Reaper, was a prolific serial-mass murderer who had escaped justice for years).
Morgan's true archenemies were Carl Buford and Rodney Harris. Buford was a child killer and pedophile who molested Derek when he was just a kid and later framed him for murders Buford himself had committed- Morgan joined the FBI to stop people like him, and years later is still disgusted and terrified of him, to the point that merely shaking Buford's hand s enough to make Morgan physically sick. Harris, meanwhile, has been Morgan's enemy since both were kids where Harris was the local bully and gang leader; while a Small Role, Big Impact, he serves as Morgan's Evil Counterpart- a kid from the streets who turned to crime and drug dealing and years later is still a street level drug dealer, while Morgan worked his butt off to earn a prominent job in law enforcement. While Morgan encourages kids to stay off the streets and get a good education, Harris encourages them to turn to crime and run drugs for him. And finally in season 8, it is revealed that Harris too was molested by Buford, but he ultimately became a Vigilante ManSerial Killer targeting men he mistakenly believed were abusing their own kids because, unlike Morgan, he let his bitterness get the best of him.
CSI mostly focused on a killer of the week formula, but an occasional seasonal Big Bad would emerge, a few of which ended up becoming arch enemy to one of the CSI team members.
In the early seasons of the show, forensically-gifted special effects artist and Evil Genius Paul Millander served as the arch enemy to team leader Gil Grissom.
Serial Killer Nate Haskell was the Arch Enemy of Ray Langston. When the two are first introduced Langston is lecturing Criminology and Haskell is a frequent "guest" speaker from his prison cell via video link, and grows to treat Ray as a Friendly Enemy (Ray, in contrast, just thinks Haskell is a despicable monster). He assists the team in capturing one of Ray's other nemesis', Mad DoctorSerial Killer Doctor Jekyll, just to get the chance to attack and nearly kill Ray. It emerges that they both share a particular gene that has been linked to a predisposition to violence, and both had violent abusive fathers, meaning Haskell is Ray's Shadow Archetype as he secretly fears they are Not So Different and Haskell is what he could become. In season 11 Haskell escapes and engages in Criminal Mind Games with him, culminating in abducting Langston's ex-wife, murdering her new husband, just to torment Ray, and in the series finale Ray snaps, and proceeds to beat and murder him.
The Daily Show: Jon Stewart and Brian Williams ironically pretend to have this relationship while actually being friends. There is nothing ironic, however, about the enmity between Jon and Tucker Carlson. Carlson has clearly never gotten over Stewart single-handedly getting Crossfire canceled, and in the process delivering a withering "Reason You Suck" Speech to Carlson, his co-host Paul Begatta, the media in general and CNN in particular. Carlson never misses an opportunity to insult or attack Stewart on air, frequently using the wording "Partisan Hack" which Stewart used when denouncing Crossfire and Stewart hasn't exactly made a secret of his opinion of Carlson.
Jon Stewart: [After calling Tucker Carlson an arrogant douche] Tucker and I can talk like this because we already have a visceral negative reaction to each other. Not an ounce of friendship or respect between us. Truly one of the only people in the world I feel that way about.
Surprisingly averted with Bill O'Reilly. Even though Fox News is The Daily Show's primary target for lampooning, with Bill and Jon regularly confronting each other in debates on the other's shows, the two seem to be actual friends in real life.
The Master (at one point the Doctor specifically refers to him /her as "my archenemy") and Davros to the Doctor. Also the Daleks, who Davros created, are the most recurring villains and were described as the Doctor's archenemies by Amy Pond in "Victory of the Daleks".
Despite the Daleks being more prominent, rivalry with the Master is nearly as personal, if not more. It helps that often they're depicted as two last of their kind with obviously different strategies of surviving.
9: The Daleks and Blon Fel-Fotch Passameer-Day Slitheen
10: The Daleks and the Master
11: The Silence and the Great Intelligence
12: Missy and Rassilon
Madame Kovarian, to the Pond/Williams family.
The Daleks as the Doctor's ultimate archnemesis was best summed up when the Eleventh Doctor took a wrench and proceeded to beat one with it;
The Doctor:YOU. ARE. MY. ENEMY! And I am YOURS! You are everything I despise! The worst thing in all creation! I've defeated you time and time again! I've defeated you! I've sent you back into the Void! I saved the whole of Reality from you! I am the Doctor— [kicks it] —and you are the Daleks!
Corakinus and the other Shadowkin to the main characters in Class (2016). They appear in four of the first season's eight episodes and have killed at least three people closely connected to the heroes.
Farscape had first Crais and then, following Crais's Hazy Feel Turn, Scorpius for John Crichton and the crew in general. But it also gave some of the other regular characters personal arch-enemies: Aeryn had Xhalax, Zhaan had Maldis, and Rygel had Durka (until he killed him and literally put his head on a stick).
Cersei Lannister has her brother Tyrion, who she fears has been destined to murder her. Tyrion doesn't truly start to hate her until she tries to have him executed for her son Joffrey's murder, though she is still second to their father for Tyrion. Cersei also has a serious rivalry with her daughter-in-law Margaery Tyrell, who she sees as the manipulative social climber she is.
Littlefinger and Varys, colleagues on the Small Council who deeply loathe one another and are playing diametrically opposed strategies for the future of the realm. While Varys at first sees Littlefinger as a Worthy Opponent, he soon acknowledges how dangerous an ambitious man can be.
Robert Baratheon and Prince Rhaegar Targaryen, both leading rival factions of the civil war that put Robert on the Iron Throne. They both are vying for the same woman, Lyanna Stark (though we only hear Robert's side of the story). The Mad King is almost an afterthought for Robert compared to his hatred for Rhaegar.
The Starks and Lannisters begin the series barely able to share a room without being drunk and it only gets worse. While initially driven by Ned's bitter opinion of their Sack of King's Landing and Jaime Lannister's Bodyguard Betrayal of King Aerys, it quickly becomes It's Personal when Joffrey has Ned executed on a whim, an action which the Lannisters did not want to do, with even Cersei wanting Ned to be sent to the Night's Watch. After that, Tywin and Tyrion realize that the remaining Starks will hunt them down.
The Night's Watch and the wildlings have raided and skirmished non-stop for so long that both have all but forgotten their true enemies, the White Walkers, who are just now returning.
The extinct Boltons were historically the Starks' chief rival for supremacy of the North. They even rebelled against the North once (a la the "Reynes of Castamere") but the Starks pardoned them after they eventually bent the knee and they were allowed to remain the second greatest house in the North. Since then, the Boltons have been forced into Teeth-Clenched Teamwork for the most part, with the Starks forcing them to outlaw their "traditions" of flaying people and Roose Bolton fighting for Ned Stark during Robert's Rebellion and supporting his Liege Lord Robb Stark during the War of the Five Kings. This ancient grudge is revived big time when Roose Bolton finds an opportunity, courtesy of Tywin Lannister, to become The Starscream and conspires to murder his Robb Stark and usurp his titles. He, along with the Freys, betrayed the Starks during the Red Wedding, with Roose personally killing Robb Stark and becoming Warden of the North and claiming the Starks' ancestral seat Winterfell as a reward. This was the first time the Boltons held the upper hand over the Starks and the only time, as in Season 6, Ramsay murders his father Roose Bolton, step-mother Walda and his newborn half-brother and the Red Wedding would later be avenged and the rivalry would come to an end with Jon Snow and his sister Sansa Stark's defeat of Ramsay in the Battle of the Bastards and with Ramsay's defeat and Karmic Death. Winterfell is restored to House Stark and House Bolton is ended forever as a threat to the Starks.
By the Season 3 premiere, Davos has definitely become this to Melisandre, though she insists she's not his enemy.
Davos: You are my enemy.
Subverted by the start of Season 6, as he still doesn't like her, per se, but he is not above asking her for help and seems to have a begrudging respect for her ability to have faith in something, even if he doesn't share it. This faith ends up paying off for a certain Lord Commander of the Night's Watch. In a huge way. However, she once again returns to full Arch-Enemy status after he discovers that Shireen was burned to death by her faith.
Brienne has said more than once that she wants to kill Stannis for murdering Renly. Stannis, however, isn't even aware of her existence, and at the time of his death, regards Brienne's grudge against him with indifferent amusement rather than dislike.
Sue Sylvester from Glee schemed constantly to bring down Will Schuster.
Sylar and Mohinder in Heroes. While there are many other contenders, with this couple there's the whole "You Killed My Father" issue. Also, since Sylar did a Heel–Face Turn, he's on the same team as all his other supposed-arch enemies. But then Mohinder had a turn of his own, so he and Sylar are still on opposite sides of the fight.
Justified: The Bennett clan are the collective archenemies of the Givens family, going all the way back to Prohibition. After years of feuding, both the Bennetts and the Givenses are down to a pair of nuclear families whose simmering grudges are kept in check only with extreme effort from the family matriarchs. Protagonist Raylan Givens has his own personal archenemy in Boyd Crowder, local crime boss and series deuteragonist.
While the enmity between protagonists and antagonists rarely become personal in the Kamen Rider franchise, there are a few notable relationships scattered throughout the franchise that could cover this trope.
Kamen Rider Black: Kotaro Minami/Kamen Rider Black and Nobuhiko Akitsuki/Shadow Moon.
Though he only appeared in two of the four season one episodes of Lexx, His Divine Shadow was this to Kai, last of the Brunnen-G. Kai's grudge against him was very personal, considering His Divine Shadow destroyed his world, rendered his race extinct, and personally stabbed Kai to death and turned him into an undead brainwashed assassin. Kai was also the person destined to kill His Divine Shadow. The conflict between them went even deeper than Kai realized since His Divine Shadow was really the last survivor of the Insect Civilization whom the Brunnen-G defeated long ago. While His Shadow's goal was to eventually Kill All Humans to achieve his rebirth, he went out of his way to wipe out the Brunnen-G personally with his own flagship.
Lost has Jacob and the Man in Black, and while Ben Linus earned plenty of enemies, his arch rivals are Charles Widmore off the island and John Locke in it.
MacGyver, the man who refuses to kill no matter what the circumstances, has Murdoc, the world's greatest professional assassin. Originally the arch nemesis of American intelligence officer Peter Thornton, Murdoc transfers this relationship to MacGyver after the latter stumbles into a confrontation between the two. Obsessed with revenge after nearly dying during this event, Murdoc returns time and time again to try and kill MacGyver, always failing but always surviving to fight another day. After trying to retire from his profession, he is forced to team up with MacGyver to rescue his sister from his former employers, and the vendetta seems to be over. However, Murdoc returns to his old life within a year, and has assigned MacGyver as a target again, rekindling the relationship. Throughout it all, Murdoc develops a healthy admiration for MacGyver's skills as the only man he was not able to defeat, while MacGyver's fear of Murdoc is unmatched by any of his other rivals.
Arguably, Adrian Monk's arch enemy is Dale "the Whale" Biederbeck. While Biederbeck is defeated by Monk in his first appearance, he appears a couple more times afterwards to taunt Monk. He provides a good counterpoint for Monk, as Biederbeck is mentally comfortable but hugely overweight to the point of being bedridden, while Monk can move around freely but is, to paraphrase Biederback, trapped in the prison of his own mind.
NCIS has terrorist/renegade Mossad agent Ari Haswari, which is this to Leroy Jethro Gibbs. He's even reffered to and appears as Posthumous Character a few times after he finally fies in season 3.
In Once Upon a Time, Regina (the Evil Queen from Snow White) is simultaneously Arch Enemy to both Snow White and her daughter Emma Swan at the beginning, but their relationship varies over the course of the show as Regina goes through the Heel–Face Revolving Door.
George and Elaine each have their own as well: Lloyd Braun and Sue Ellen Mischke, respectively. In these cases, the rivals seem unaware that George and Elaine aren't their friends and are actively rooting or plotting against them.
Lampshaded in Sherlock when Mycroft and Sherlock refers to each other as such, and this conversation happens:
Dr. John Watson: People don't have archenemies. Sherlock Holmes: What? Dr. John Watson: In real life. People don't have archenemies. Sherlock Holmes: That sounds a bit dull. So what do people have in their REAL lives? Dr. John Watson: Friends, people they like, people they don't like, boyfriends, girlfriends... Sherlock Holmes: Like I said, dull.
JimMoriarty is eventually revealed and solidly placed in the position of Sherlock's arch-enemy.
In the third season, Charles Augustus Magnussen becomes Sherlock's new Arch Enemy, and arouses Sherlock's genuine and implacable hatred in a way that Moriarty didn't.
Clark Kent has Lex Luthor and Zod. Lex is his one-time best friend turned bitter professional and romantic rival, delving into supervillainy at the same time that Clark is hitting his stride as a superhero; the entire series is essentially spent building up to a confrontation between the two. Zod is a former friend and ally of Clark's father, Jor-El, with Clark inheriting the grudge. Given that Zod corrupted Brainiac, created Doomsday, and blew up Krypton he's responsible for almost every horrible thing that's happened in Clark's life, and his multi-seasonal attempts to Take Over the World only turn it more personal.
Lex's father, Lionel Luthor, is the archenemy of Clark's adoptive father, Jonathan Kent. The two spend their entire time on the show at war for the souls of their sons, while doing everything they can to tear one another down. Lionel (and his alternate universe doppelganger) would also place an impressive third on Clark's enemy.
Stargate Atlantis: Acastus Kolya was this to Sheppard. They only met a handful of times over the course of the series, but there was more animosity between them than Sheppard had with any other villain on the show.
That being said, the rogue Wraith/human hybrid Michael could be seen as an Arch Enemy to the team as a whole (and particularly Tayla in his later appearances).
Even the way the rivalry between Sheppard and Kolya ended is straight out of a western: a quick draw that Kolya loses.
Very bizarrely subverted in an episode of Terminator: The Sarah Connor Chronicles. Derek is introduced to a man who he is told is a pro-Machine human agent sent back from the future, but doesn't recognise him. It is revealed that in the timeline Jesse and the other man come from, he tortured Derek for weeks and became his personal Arch-Enemy. The fact that Derek has no memory of any of this reveals that Derek and Jesse actually come from different potential futures.
In White Collar, MatthewKeller is arguably this to all four main characters, Neal in particular due to being his Evil Counterpart, as both the most-recurring villain of the series and one of the most dangerous. His first episode makes it clear that he and Neal used to have a sort of rivalry over Neal's girlfriend Kate, and later episodes only ramp up his animosity with the other characters:
Season 2 sees him having Peter kidnapped and almost murdered, while Keller himself escapes from prison in the chaos;
He comes back again in Season 3 to attempt to force Neal and Mozzie to share their treasure with him, and when they refuse, he proceeds to menace them both and almost kill Neal, succeed in killing one of their fence friends, and kidnap Elizabeth and hold her hostage to force Neal, Mozzie, and Peter to help him steal the treasure, and betrays and almost kills Neal again before the operation is over. This proves traumatic enough for all of them that it takes a few episodes for them to get over it.
By the time he reappears in Season 6, the others hate Keller so much that Neal is barely civil to him despite being in an Enemy Mine situation for an undercover operation during the whole season, and Peter's first reaction to finding out about his presence is to want to immediately arrest him and put an end to the operation. In the series finale, Keller again betrays and (apparently) succeeds in killing Neal, and is finally killed by Peter.