Zero and Suzaku in Code Geass, though it's another case of moral ambiguity/ideological antagonism. They eventually get over their issues and honestly work together.
For Lelouch himself, both Charles and Schneizel follow this straight. And maddeningly enough, he has the misfortune to get sold out to both to them after they've already caused him enough grief for several lifetimes.
In Naruto it's a little hard to decide, but mostly earlier friends turn into enemies, such as Orochimaru and Jiraiya, or even Naruto and Sasuke (though, the latter is in a quite difficult situation). Also Sasuke considered Itachi his arch-nemesis for a long time until learing the truth now turns to Naruto and the leaf village. Tobi and the Akatsuki to Naruto and his allies is perhaps the primary one.
The Second Tsuchikage and Second Mizukage had a long-standing grudge, which eventually led to their final battle where both perished.
Madara Uchiha to everyone. Literally speaking, everything that went wrong in the 'verse that would eventually lead to the plot can be blamed on him. The fact that his ultimate goal is to destroy free will is what causes the entire ninja world, who has been in almost a constant state of war for the last century, to unite and stand against him, speaks volumes about what kind of man Madara Uchiha is.
In Detective Conan Shinichi's rival might be considered Kaito Kid, wherefore his nemesis is 'the organization' with his most present member Gin.
Char Aznable and Amuro Ray are perhaps the most iconic throughout the Gundam fandom. What started out as a professional military rivalry, later turned deeply personal after the death of Char's assistant Lahlah Sune, whom both men loved, when she attempted to intervene in a battle between the two of them. In the end their rivalry leads to a Mutual Kill in Char's Counterattack.
Zeta Gundam: While Paptimus Scirocco might be the Big Bad and Yazan Gable the resident Hero Killer, it's Titans' Ace Pilot and former Jerk Jock Jerid Messa who is protagonist Kamille Bidan's archetypal archnemesis. The two of them essentially go back and forth on killing one another's friends and loved ones (Jerid kills Kamille's mother, girlfriend, and one of his friends; Kamille Jerid's crush, his Big Brother Mentor, his girlfriend, and a truckload of his subordinates), until the rivalry turns deeply personal. By the end all that Jerid in particular is living for is his dream of seeing Kamille's head on a pike.
Contrasting them is the equally twisted relationship between Roux Louka, Judau's right-hand girl, and Haman's treacherous subordinate Glemmy Toto. From shortly after his first appearance, Glemmy becomes obsessed with Roux, stalking her wherever she goes, and doing his best to gain control over her, while Roux alternately manipulates and defies him, using him for information one moment, then trying to kill him the next. Glemmy, in return, begins to subvert Roux's expectations, using her attempts at manipulating him in order to gain advantage over her crewmates. In the end, it's Roux who kills Glemmy, who accepts it with grace.
In G Gundam the Canadian fighter Andrew Graham considers the Russian Fighter Argo Gulski his nemesis because he blames him for his wife's death. Also Domo Kasshu's and Master Asia's difficult relationship might be some kind of arch nemesis.
Rau Le Creuset of Gundam SEED is the Arch Enemy of Earth Forces' Ace Pilot Mu La Flaga. Their rivalry goes back to early in the war, and is extremely personal, even before The Reveal that Le Creuset is a dying clone of La Flaga's father. Early in-show, the Le Creuset team, and especially Athrun Zala are set up as the Archenemies of the Archangel, and Kira Yamato in particular. By the end they have pulled a Heel-Face Turn and Le Creuset takes their position as Kira' Arch Enemy.
In Gundam 00 Graham Acre considers Setsuna F. Seiei and his Gundam Exia his arch nemesis for their crimes (though he calls it a special kind of love...) But he's really more The Rival in that regard. A better fit for Setsuna's archnemesis is psychopathicBlood Knight Ali Al-Saachez for destroying not just his life, but also killing off Lockon Stratos' family as well.
In Gundam AGEDecilGalette is Flit Asuno's Arch Enemy starting late in Season 1 and continuing into Season 2. Flit might hate all of the Vagan with a passion, but it takes Decil's return to send him into a murderous enough rage that he sorties himself, while Decil is living for revenge after Flit humiliated him in Season 1's climax.
Gauron and Sousuke in Full Metal Panic!!. The feeling tends to be a lot more one-sided on Gauron's part, as he makes it his personal mission to put himself in a position where he can constantly antagonize Sousuke. It's so bad, it's to the point where it's suggested that he actually stalks and watches Sousuke, and purposefully chooses missions that would put him at odds with Sousuke, allowing them to meet again. Sousuke, on the other hand, finds Gauron insanely annoying, and would really love nothing more than for the guy to just go die somewhere and leave him alone.
The original enmity was between Ash and Gary, with Gary rubbing his superiority in Ash's nose and Ash going crazy as he tried to catch up. They eventually buried the hatchet when they finally battled at the Johto League. Although Ash beat Gary, Defeat Means Friendship and they became friendly rivals afterward.
Ash later developed a more bitter rivalry with Paul over their methods of training Pokemon. A combination of Break the Haughty on Paul's part and Ash eventually beating him in the Sinnoh League again led them to bury the hatchet, and while they didn't exactly become friends afterward, they did develop a new respect for one another.
Galactic Commander Saturn's Toxicroak to Brock's Croagunk.
Team Rocket's Meowth feels this way towards every Persian he comes across.
Ranma of Ranma ˝ had Happosai (outright stated to be Ranma Arch Enemy in the datebook)
Harima the delinquent in School Rumble had Hanai the class president. Harima's love rival is Karasuma. Eri & Yakumo too.
In Bleach, Ishida originally had Mad Scientist Mayuri Kurotsuchi as his nemesis, although now that they're on the same side, Ishida seems to come across as the Butt Monkey victim of Mayuri's sociopathic humor. The protagonist always had at least one mayor rival at each point in the story (first Ishida, followed by Renji, Kenpachi, Byakuya, Grimmjow and Ulquiorra).
We also have The Grand Fisher, the first major antagonist. He went down quickly, but at the time he was Ichigo's archnemesis. He killed Ichigo's mother, the event that changed his life completely. It is because of him that he changes his motivation for fighting hollows, taking it much more seriously. Grand Fisher seems to acknowledge this, seeking out Ichigo as soon as he returns to the human world.
GUN×SWORD - Justifying the Designated Girl Fight, Carmen 99 and Fasalina are arch enemies. Fasalina is responsible for destroying Carmen's hometown, causing the death of Carmen's best friend from childhood, and Carmen ends up blamed for Fasalina's actions and shunned and exiled from her town.
Griffith to Guts in Berserk. Griffith claims not to care much one way or the other about Guts after Griffith ascends to the God Hand, but his actions give it the lie. The Skull Knight also has one in Void, the leader of the Godhand. Zodd considers the Skull Knight his, although Skully doesn't seem to take their rivalry quite as personally, considering him a Worthy Foe but not much more.
In Durarara!!, while almost anyone can set off Shizuo's Hair-Trigger Temper, just seeing Izaya (and sometimes just hearing his name) is enough to send Shizuo into a homicidal rage. Meanwhile, Izaya screws around with everyone's lives, but that's just because he likes seeing how they'll react; Shizuo's the only person in the world he actually hates.
In Soul Eater, Medusa is the Arch Enemy of Maka, on Maka's part due to Medusa's connection to the black blood and Crona, and on Medusa's side thanks to Maka's Soul Perception. Asura is this Lord Death and his DWMA Students, Staff, and many of their allies, including The Heroine Maka and her friends.
In HunterxHunter, Hisoka is this to Gon. He has an unnerving obssession with Gon and anticipates the day he "ripens to his full potential" just so he can have the chance to fight and possibly kill him. Gon also had a desire to fight Hisoka earlier in the series for hurting his pride.
Chrollo Lucifer is Kurapica's Arch Enemy, due to being the leader of the group responsible for killing his clan.
The one person Killua hates and fears more than anyone else is his own older brother Illumi. Illumi is psychotically devoted to Killua and has done some truly awful things to him and to others because of it.
In Jojos Bizarre Adventure, Dio Brando is this to Jonathan Joestar in Part I and arguably to the Joestar bloodline for the rest of the series. Even after he is permanently slain in Part III, the mess he left behind causes trouble for Jonathan's descendants.
One Piece: While for the most part subverted due to the Arc Villain formula, there have been exceptions.
Vivi, who's practically an angel, hates Crocodile with such a passion that he was the only person she has ever expressed the desire to kill. Then again, he's more than earned her hatred, and that of all of Alabasta for that matter.
Luffy, who may be one of the most forgiving people in the series, has aided several of his enemies, including Buggy and Mr. 3, but immediately had his live-and-let-live tendencies shut down the very moment he met Crocodile again while in Impel Down. Luffy would have never let him out of his cell had it not been for Ivankov.
Crocodile himself has or had one in Whitebeard.
Akainu to Luffy and the remnants of the Whitebeard Pirates for killing Ace.
Blackbeard to the Whitebeard Pirates for betraying them and killing nakama, then orchestrating the Marineford war for his own personal gain, thereby indirectly causing Ace's death, and then personally killing (with the help of his crew) Whitebeard himself.
Law and Doflamingo. To the point that Law is willing to do anything to ensure Doflamingo's death, even at the cost of his own life. Doflamingo, after the last two arcs, has come to hate him just as deeply.
Doflamingo is also a personal enemy of the Riku royal family, the previous rulers of Dressrosa. Let's put it this way: what Crocodile did to Vivi and Alabasta was nothing compared to what Doflamingo did to the Riku family and Dressrosa as a whole.
Subverted in Suite Pretty Cure ♪. It seems that Mephisto is the Arch-Enemy to Aphrodite, but it later turns out that Mephisto actually is the king of Major Land, Aphrodite's husband and he's brainwashed by the real Big Bad Noise.
In Doki Doki Pretty Cure, Regina is this to Cure Heart. With a lots of Foe Yay. Sadly, both actually love each other (in a platonic way), and Cure Heart wants to save Regina because she's brainwashed by her father King Jikochuu, Regina is still a villain.
In Dragon Ball Z, Vegeta is this to Goku until his Heel-Face Turn. Before the transition to the "Z" era, Piccolo was a strong contender.
In Fairy Tail, the demon Deliora ruined Gray's life by wiping out his hometown, killing his parents, and forcing his mentor/surrogate mother Ur to sacrifice herself to defeat him. When Gray encounters Deliora again when the latter is possessing the corpse of Gray's father Silver Fullbuster, Gray loses it.
In Fist of the North Star, main character Kenshiro's arch-nemeses were his adoptive older brothers Jagi and Raoh, who were both bitter over Ken being chosen as the sole successor to Hokuto Shinken over them (and Toki, but he didn't really mind). Even though Raoh was the bigger rival in the story, Jagi gets special mention because he purposefully set out to ruin Ken's life and is personally responsible for the good majority of the plot. In the early part of the series, Shin fufilled this role, as he was the one who took Ken's fiance Yuria away from him and gave him his distinctive seven scars on his chest, though it could be considered subverted in the meta-sense because he was goaded into doing this by Jagi.