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, Kakashi and Obito, or even Naruto and Sasuke (though, the latter is in a quite difficult situation). Sasuke also considered Itachi his arch-nemesis for a long time until learning the truth behind the latter's killing of the Itachi clan, leading to the former turning his ire to Naruto and the Hidden Leaf Village, especially Danzo Shimura, who ordered the massacre in the first place. The Hidden Leaf 11 as a whole consider Sasuke to be their nemesis. Tobi and the Akatsuki to Naruto and his allies is perhaps the primary one. Pain and Konan to Jiraya, their former sensei. After Jiraya's death, Naruto devoloped a vendetta against Pain. Zetsu to both the Senju and Uchiha clans, having caused and manipulated the conflict between them for centuries. Deidara to Gaara, who sacrificed himself (temporarily) to save his village from Deidara, and Sai, a rival art-based jutsu user. Kisame Hoshigaki to Might Guy, although Guy keeps forgetting him, and Killer Bee, whom Kisame tried to capture. Hidan and Kakazu to Shikamaru Nara, Choji Akimichi, and Ino Yamanaka, because Hidan killed their msensei, Asuma. Sasori to Sakura Haruno, Kankuro, and Chiyo.
Orochimaru seems to make it his life's work to earn the hatred of a significant portion of the cast, from Anko Mitarashi and Hiruzen Sarutobi, to Jiraiya and Tsunade, and is probably this to the Leaf as a whole given his goal is to wipe it from the face of the Earth and that he topped its Most Wanted List. Naruto probably feels this way about him too, since even though Oro hasn't done that much to him personally, he doesn't forgive him for murdering the 3rd or corrupting Sasuke.
Kabuto Yakushi is arguably this to Sakura Haruno, due to the fact that they are both medical ninja, making Kabuto and Evil Counterpart, and have confronted each other multiple times.
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. Madara himself considers Hashirama Senju and Tobirama Senju to be his archenemies.
Gaara's was his father, at least until he died. Ironically, Gaara wasn't the one who killed him (though, there is no doubt he probably wished he was at the time) — Orochimaru was. By the time they finally had their fight (via Edo Tensei), Gaara had already forgiven him.
Kurama, the Nine-Tails, considers Minato Namikaze to be his nemesis because Minato sealed him away.
Kaguya Otsuksuki to her two children, Hagoromo and Hamura.
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 Mobile Suit Gundam: 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.
Red and Giovanni in Pokémon Special are the most important. Also, Yellow and Lance, with a dash of Evil Counterpart for good measure. The only reason Petrel managed to get the drop on Lance in the HGSS arc is because he took the appearance of the person who disturbs Lance the most — Yellow.
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.
In both the games and the anime, Groudon and Kyogre are this to each other.
Ranma of Ranma ˝ had Happosai (outright stated to be Ranma's 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.
Ichigo also had one in Grimmjow, though that was mostly by the latter's insistence. Ichigo himself seemed to have mixed feelings on the matter.
Nel and Nnoitra have incompatible philosophies. She believes in never fighting without reason and avoiding it if possible, he believes fighting itself is a reason to fight. He hates women, especially women in positions of authority over him, as Nel used to be and Harribel is. Nel dismisses his philosophy and refuses to see him as a warrior. As a result, she feels he needs to be watched over to prevent himself from getting into more trouble than he can handle. Eventually, he rips off her fraccions' masks, smashes her mask and throws her out of Las Noches.
Kimblee to Scar in Fullmetal Alchemist, as well as Hohenheim to Father. Wrath also becomes this to both Greed and Scar.
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 Opponent 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.
Izaya isn't all that fond of Nakura either.
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.
The two organizations they work for, Hellsing and Iscariot respectively, are arch enemies andrivals.
In Hunter x Hunter, Hisoka is the closest thing Gon has to one. 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, and Gon also had a desire to fight Hisoka earlier in the series for hurting his pride. Neferpitou becomes this to Gon during the Chimera Ants arc, after murdering Kite and using his corpse as a training dummy for other Chimera Ants.
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.
Aokiji also has major beef with Akainu, though more from an ideological standpoint. However, their enmity and rivalry did not become so prevalent until Sengoku retired and both competed for the position of Fleet Admiral. Aokiji, normally indifferent, even somewhat adverse to the idea of inheriting such a tedious job, was completely against Akainu becoming Fleet Admiral. This feud between them, the first ever major conflict between admirals, became so heated that it eventually degenerated to a Duel to the Death on the barren island of Punk Hazard, with the winner becoming Fleet Admiral. The victor wound up being Akainu; but for all his ruthlessness, not even he could ignore the decades he and Aokiji served together as comrades and ended up sparing his opponent's life. However, Aokiji despised the idea of serving under his rival so much that he ditched the Marines and gave up his position as an admiral.
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.
Though he was unaware of it at the time, Doflamingo's own brother Rocinante (aka Corazon, Law's idol) held great enmity for him. While the exact circumstances are unknown, Doflamingo would end up murdering him, which would earn him Law's hatred.
Chapter 764 implies that those bearing the D. initial are this to the World Nobles — to the point that they're nicknamed "God's Natural Enemy". In hindsight, this wasn't really surprising at all, considering how much practically every D. in the series constantly chafed against the World Government and the World Nobles in particular. Law's conflict with Doflamingo just happened to be another example, albeit one we weren't entirely aware of first, while Luffy's actions speak for themselves.
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.
Most of the villains being Arc Villains, the series doesn't have a lot of this, but Frieza is a strong contender for Goku & Vegeta. Frieza is responsible for the extinction of their species and homeworld, pushed Goku to such a level of Unstoppable Rage when he's normally such a Nice Guy by killing Krillin and kept Vegeta as his servant for years after killing his father. On the flipside, Frieza absolutely despises Goku for humiliating him in their battle on Namek. And it was Trunks, Vegeta's son, that ultimately finishes him off in the following arc. In Resurrection F Frieza outright says that he can't stand to be in an universe where Goku is alive.
In the original series there is Piccolo, both of them. King Piccolo murdered Krillin and Master Roshi, pushing Goku into a giant Roaring Rampage of Revenge. When King Piccolo was reborn into the current Piccolo, he dedicated his life to killing Goku. They remained bitter enemies until Raditz's arrival.
Cell to both Gohan and Future Trunks.
The Androids from the future would be a better example for Future Trunks, seeing as how they killed almost everyone in his timeline.
The Supreme Kai to Babidi and to a lesser extent Majin Buu. The Supreme Kai personally kills Babidi's father and makes it is life mission to stop Buu since Buu killed all his friends and mentors.
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.
One can make an argument that Acnologia is the arch-enemy of the Dragon Slayers, seeing as he's the one responsible for the deaths of their adoptive parents. A more concrete example would be Natsu, seeing as Acnologia killed Igneel right in front of him — the rest went out in a more Ascend to a Higher Plane of Existence kind of way.
Black Mage Zeref and Mavis Vermillion. They were once teacher and student, then (for an all too brief moment) lovers. Now they are sworn enemies, with Zeref vowing to wipe out humanity and Mavis in turn vowing to destroy him.
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.