Cartman and Kyle from South Park. It was mostly onesided on Cartman's side earlier on, but the two snowballed into mortal enemies somewhere in the fifth season, and has jumped up a notch with every subsequent season.
Also, Scott Tenorman to his half-brother Cartman.
Wendy Testaburger probably also considers Cartman her archenemy
The Monarch from The Venture Bros. wants to be Doctor Venture's Arch Enemy, but Doctor Venture doesn't consider him a threat.
Hank: What's your problem with our dad anyway?!
The Monarch: (awkwardly) Well, I- he- he's my nemesis. My Arch Enemy.
Dean: I don't think pop thinks you're his Arch Enemy.
The Monarch: Come on, I'm sure the walls of the Venture Compound are practically caked with the lingering curses of the Monarch's name.
Dean: Uhh, no. I've never even heard him mention you.
Hank: Yeah, I always thought Baron Underbheit was dad's arch-enemy.
The Monarch: (astounded) Underbheit!?! Why, that dime-store Doctor Doom isn't fit to — just you wait till your father calls me back!!
Initially, at least. The Monarch quickly revealed himself to be a Not-So-Harmless Villain and an actual Diabolical Mastermind over the course of the first and second seasons. He was also involved in a vicious feud with Underbheit over who could kill Venture (before they decided upon a Villain Team-Up). By the end of the second season, Rusty wasn't willing to bury the hatchet, even though the Monarch was (though it's all back to the same routine by the later season).
Phantom Limb actually became this to The Monarch and Dr. Venture towards the end of season 1, but underwent Villain Decay after his defeat.
Underbheit, on the other hand, has made very rare appearances now and again, until La Résistance deposed him of his throne. He hasn't been, and (according to DVD Commentary) probably won't be, heard from again.
He has been. About halfway through Season 4, he teamed up with Phantom Limb and Professor Impossible.
And, as of season three's "Home Is Where The Hate Is", Dr. Venture's new Arch Enemy was Sergeant Hatred, who was assigned by the Guild of Calamitous Intent. His actual goal is to be as nice to Dr. Venture as possible, so as to annoy the Monarch and pay him back for his henchmen robbing him. However Hatred eventually becomes the new Venture Bodyguard and Monarch gets himself a free pass to antagonize Venture once again.
The Venture Bros. has an interesting system wherein archrivalries are assigned, not chosen, by the OSI and Guild of Calamitous Intent (hence the coinage of a term: 'arching' to describe the act of being an archnemesis). In season 3, it's revealed that a reason for Dr. Mrs. The Monarch's attraction to her husband is that The Monarch's archrivalry of Venture doesn't stem from this stodgy system, but actual, living, breathing, unfounded hatred.
Captain Hero on Drawn Together belatedly realized that the random homeless guy they ran over was "My... ARCH... NEMESIS!"
As an added, subtlety that would only be found on this show, his goal is not Captain Hero's destruction, but to get him to wash his balls. And has succeeded multiple times.
David Xanatos to Goliath in Gargoyles. Though Xanatos fits best from a series standpoint, from a personal standpoint Demona, the Archmage, and Thailog might also want to claim that spot. From the same series, Tony Dracon is Elisa Maza's Arch Enemy (though he's not much of a threat to the Gargoyles themselves), while the line of Hunters consider themselves Demona's collective Arch Enemy (though in truth hers is probably MacBeth).
We never learn the details, but as per some kind of feud stretching to before the series, Dr. Quest and Dr. Zin seem to both hate and grudgingly respect one another's guts in Jonny Quest.
For a very brief time (about two episodes), SpongeBob himself had Bubble Bass; a slimy, devious and notoriously hard-to-please connoisseur who served as a Foil to SpongeBob, the workaholic, perfectionist fry cook.
Sometimes: SpongeBob and Plankton.
On The Simpsons, Bart Simpson has two archenemies (and both are completely mutual): Principal Seymour Skinner and Sideshow Bob. With Skinner, it's an arch-rival type of relationship: Bart's efforts to bring chaos to the school vs. Skinner trying to bring order. With Bob, it's a much more serious and deadly relationship (like arch-nemeses): Bob trying to kill Bart vs. Bart trying to put Bob back in jail.
In the movies-within-the-show, McBain's archrival is Mendoza. Mendoza!!!!!!!
Even Maggie has the the one-eyebrowed Baby Gerald.
Homer's archenemies are Mr. Burns, Ned Flanders, Frank Grimes, and Patty and Selma.
Although with Grimes it's mainly one sided, Homer seemed to actually quite like him. It's also one-sided with Homer and Flanders, since Flanders considers him to be a close friend.
On a personal level the Brotherhood (and specifically, the Brain) is Beast Boy's Arch Enemy. He was a member of the team that opposed them before he joined the Titans, and his past with them is what draws the other Titans into the conflict. Interestingly, this is a rare case of the hero taking it more seriously than the villain- the Brain considers Mento, the Doom Patrol's leader, to be his Arch Enemy, and regards Beast Boy as little more than an irritant. Yet, thanks to Beast Boy (and by extension, the Titans), the Brain considers ALL kid heroes his archenemies.
Avatar: The Last Airbender: Prince Zuko is the Arch Enemy of both Aang and Zhao for the first season. In the second season, Zhao dies and his evil sister Azula becomes Zuko's new Arch Enemy. Aang himself himself has Fire Lord Ozai, though mostly from the fact that the two of them are destined to have an ultimate showdown, and everyone who wants to capture Aang intend to present him to the Fire Lord.
Dr. Drakken is Kim Possible's self-proclaimed arch foe, although Shego is her foil and the one she actually locks fists with. More specifically Drakken is an arch foe and Shego is the nemesis. It's like a lovetriangle...
Ron's Arch Foe is Monkey Fist, but that's mostly because Fist is the only one who recognizes Ron's name; that and the "Mystical Monkey Power."
And yet Luthor has survived direct dealings with both of them while even as both suffered grievous defeat. He was fused with Brainiac and personally put away Darkseid with the Anti-Life Equation who, it should be added, was even fused with Brainiac at the time.
Of all three though, Darkseid is easily Superman's most hated enemy. To put it in perspective, he is the only villain in the DCAU that Superman has no problem with killing, and actually WANTS to kill.
The only one who was able to actually BREAK Superman was indeed Lex Luthor. That WAS the Justice Lord version, but he was still really, really close to it in the main DCAU also.
Brainiac seems to consider Superman's father Jor-El to be his archfoe even though Jor-El is long dead. Brainiac never forgot that Jor-El opposed him to the very end and nearly destroyed him. For someone who supposedly lacks emotions, Brainiac took great pleasure in making "the son of Jor-El" kneel before him when he went back in time to kill teenage Clark. When Brainiac and Superman encounter each other on Apokolips, Brainiac claims that Superman has no hope of stopping him since his father couldn't.
The Tick: Oh, for crying out loud... Brainchild: So, at last, The Tick, my arch enemy, sits helpless before me! The Tick: 'Arch enemy'? You must be joking, Charles. I mean, I don't like you and everything but if I ever do have an arch enemy it's not going to be some creepy little brat with a glass head.
Carter Pewterschmidt, James Woods, and Ernie the Giant Chicken to Peter Griffin on Family Guy.
Stewie to his half-brother Bertram.
Glen Quagmire to Brian.
For some reason or another, In Spider-Man's cartoon from the 80s, his arch enemy was Doctor Doom. While the two have had s few historic encounters with one another in the comics, Doom is otherwise always depicted as an enemy of the Fantastic Four above all else.
Using similar logic to the Kim Possible example, in DuckTales Scrooge McDuck's arch rival is Flintheart Glomgold, but his most threatening nemesis is Magica de Spell. If you have to have an obsessive, implacable arch nemesis, might as well be a sexy sorceress.
Freakazoid! is another example of a series where the Arch Enemy and the most dangerous enemy are not one and the same. The Lobe is the Arch Enemy, but Freakazoid's most genuinely threatening foe is Corrupt Corporate Executive Armando Gutierrez.
In Spider-Man: The Animated Series all of the three traditional Spider-Man arch-enemies mentioned in the comic books section above are present, but the Green Goblin and Venom are both far less prominent, and Dr Octopus is downgraded to a B-list villain. Instead, the Kingpin serves as Spidey's main enemy.
Ben 10 has several characters possessing their own ach-enemies:
Ben Tennyson's most well known nemesis is Vilgax, who also used to be Max's. Kevin 11 and Dr Animo were considered as such as well in the original show, but by the sequels, Kevin made a Heel-Face Turn and Animo became a relatively minor villain due to new more dangerous villains showing up. Finally, Albedo arguably counts due to being Ben's Evil Counterpart.
Gwen Tennyson has Hex and Charmcaster (until Heel-Face Turn), both being directly linked to her connection to magic. While both of them did fight Ben, Max or Kevin, it's pretty clear their grudge toward Gwen is much more personnal.
While Ragnarock appeared for only one episode, his link to Kevin's origin is enough to qualify as Kevin's Arch-enemy.
Adventures of the Galaxy Rangers had a pretty good sized Rogues Gallery for an 80's cartoon, but two out of the four definitely had archenemies. The Queen of the Crowns had Zachary as an Arch Enemy while Killbane and Shane had a long, nasty history that only got worse after most of the Supertroopers went renegade.
The Huntsman is Jake's Arch Enemy in American Dragon Jake Long. Interestingly, he is only #4 on the Dragons' list of the top thirteen threats to the magical world. The #1 threat, the Dark Dragon, is Jake's Grandpa's Arch Enemy.
Often played for laughs in Darkwing Duck, where Darkwing would frequently refer as his Arch Enemy/archrival/archnemesis any villain he would have met at least twice. Even a guys whose only power was to have a bunch of penguins at his service to commit robberies was once referred as his Arch Enemy. However, from a serious point of view, at least three antagonists can be seen as his archenemies:
Megavolt initially owned the role, being both the first villain Darkwing chronogically faced (at least in his more "realistic" backstory), a personnal ennemy and one of the most recurring antagonists. He gradually lost the post to Negaduck however, to the point the recent comic sequel has him retired.
Negaduck is mostly considered as the main Arch Enemy, due to both being an evil counterpart of the hero, the leader of the Fearsome Five and the most popular vilain amongst fans.
Taurus Bulba, while having only two appearances in the animated series, was the first villain seen onscreen, as well as one of the most fearsome and competent. His link to Gazolyne makes him a personnal villain. As such, he can be considered, in a way, as an Arch-Enemy
Numbuh 1 also had Father who is his Evil Uncle and Chad aka Numbuh 274, the former leader of the KND who hated him because Numbuh 1 was considered to be a better operative than him (which was eventually proven true). Numbuh 5 has her sister Cree, Numbuh 2, the Common Cold, and Numbuh 4, the Toilenator.
Skysurfer One and Cybron from Skysurfer Strike Force. On the one hand, Cybron is responsible for stealing a an experimental computer brain from an A.I. lab, then blowing it to kingdom come, letting Sky's father (who Cybron may have also killed) take the blame. On the other hand, Skysurfer One and the other Skysurfers routinely stop his attempts to Take Over the World.
In the Futurama movie Into the Wild Green Yonder, the Encyclopod and the Dark One. Their species have been archenemies since time immemorial: the Encyclopods wish to preserve all life and the Dark Ones want to destroy it. After the last Dark One is killed, the last Encyclopod reluctantly decides to preserve its ancient enemy's genetic code. Fortunately, the last Dark One's remains are immediately wholly destroyed making this impossible so the Encyclopod doesn't have to carry its worst enemy on its back.
Diamond Tiara and (occasionally) Silver Spoon to the Cutie Mark Crusaders (but mostly Apple Bloom). Diamond Tiara does not like Apple Bloom one bit and all of her appearances show her trying to hurt and humiliate the "blank-flank".
In almost all incarnations of the Teenage Mutant Ninja Turtles franchise, the Shredder has always been at odds with the Turtles and Splinter, and always at a deeply personal level. However, the one who arguably displays this trait strongest is the Utrom Shredder of the 2003 TMNT cartoon series. His hatred of them was so deep that it culminated into the climax of the Turtles Forever movie, where he was willing to destroy the multiverse to destroy all turtles, even at the cost of his own life and that of his daughter Karai, the only other living being that he honestly cares about.
In Wakfu, Qilby the Traitor has deeply personal conflicts with the people who were once his closest friends and companions namely the other original Eliatropes and their Dragon siblings. Those who still remember him loathe him utterly, while those who have forgotten are quickly reminded of why they hated him so much.