The Master (at one point the Doctor specifically refers to him /her as "my Arch Enemy") and Davros to the Doctor. Also the Daleks, who Davros created, are the most recurring villains and were described as the Doctor's Arch Enemy by Amy Pond in "Victory of the Daleks".
Despite 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.
In "The Magician's Apprentice", when the Doctor refers to Davros as his archenemy, the Master is visibly jealous.
Each Doctor appears to have a personal rival...
1: The Daleks and the Monk
2: The Cybermen, the Ice Warriors and the Great Intelligence
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/The Master
Madame Kovarian to the Pond/Williams family
The Daleks as Doctor's ultimate archnemesis was best summed up when the Eleventh Doctor took a wrench and proceeded to beat one with it;
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!
Considering the events of "The Name of the Doctor" the Great Intelligence could be considered this.
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. She 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 considers itself the archnemesis of Buffy, but... yeah... She doesn't take them serious either.
Big mistake. Their leader, Warren managed to screw with Buffy, mentally and emotionally, more than any other villain, save Angelus.
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.
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]
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) and Angel's arch-enemy is either Wolfram and Hart (the law firm that employs Lindsey for a good chunk of the show and several other lawyers who oppose Angel over the course of the first three seasons and the Pre-Jasmine portion of season 4) or The Wolf, The Ram, and The Hart (unseen Greater Scope Villains who are the senior partners of Wolfram and Hart and the demons behind both the season 2 finale Sinister Ministers and season 5 big bads The Circle of the Black Thorn).
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.
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.
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.
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.
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.
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.
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.
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.
Lampshaded in Sherlock when Mycroft and Sherlock refers to each other as such, and this conversation happens:
Dr. John Watson: People don't have archememies.
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.
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.
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.
Kamen Rider Gaim: Kouta Kazuaraba/Kamen Rider Gaim and Mitsuzane Kureshima/Kamen Rider Ryugen.
On Criminal Minds Frank Breitkopf was Jason Gideon's Arch Enemy (and before Frank, the bomber who killed Gideons 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).
Morgans 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 commited- 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 Morgans enemy since both were kids where Harris was the local bully and gang leader; while a Small Role, Big Impact, he serves as Morgans' 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 promient 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.
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).
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.
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.
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 Givens' 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.
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.
The X-Files has The Smoking Man (acting as a proxy to the Syndicate as a whole) to Mulder and Scully. Kyrcek also fit due to his more on-hand approach against Mulder.
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 his 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.