Soon I Will Be Invincible: After defeating Doctor Impossible and discovering her ties to him, Fatale considers this might make him her Arch Enemy, but is it worth it?
I suppose I ought to vow to oppose him forever; make a nemesis issue of it. But it's a little beside the point. ... with CoreFire back, he's already got a nemesis. I'd have to be a co-nemesis. ... I'm not 100 percent sure what I'm supposed to angry about, if we come down to it.
Hakeswill in Sharpe seems to dedicate himself expressly to making Sharpe's life miserable - he has him flogged in pre-canon, he has Harper flogged after figuring out how close he and Sharpe are, he attempts to rape and eventually murders Sharpe's wife... Sharpe tries and fails to kill him in repeated inventive ways before he finally gets to give him the coup de grace at his execution.
Once Hakeswill had been written out of the novels, this role was arguably taken over by Major Ducos, a French spymaster who is constantly coming up with schemes to hamstring, if not outright stop, the British war effort, as well as to humiliate, discredit and kill Sharpe in the process.
Lex Luthor (or the Guy who took the name-Like he cared about a copyright infringment suit) thought he was in Tales of an Mazing Girl. He was more of a minor threat.
The Duke of Wellington and Napoleon Bonaparte, whom some like to see as historical archenemies, even though they probably did not see themselves that way, show up in the series.
John Devil, from the eponymous novel by Paul Feval is the Arch Enemy of Inspector Gregory Temple, as well as being one of the earliest super-villains, anticipating the aforementioned Moriarty with about 30 years.
Carl Peterson was the Arch Enemy of Bulldog Drummond, until he was killed in the fourth novel. Afterwards, Carl's mistress, Irma took over his role.
Ballmeyer, aka Detective Frederic Larsan is the Arch Enemy of Joseph Rouletabille as well as the latter's biological father.
Franny Roote in Reginald Hill's Dalziel and Pascoe novels: although Hill dispatches Roote to jail in the series' second novel, An Advancement of Learning, he resurrects him as Pascoe's obsession in Arms and the Women, Dialogues of the Dead, and Death's Jest-Book, then finally elevates him to true Arch Enemy status in A Cure for All Diseases.
In The Wheel of Time the Forsaken Ishamael Later Moridin clearly views himself as this to Rand Al'Thor. Considering that he's The Verse's version of The Antichrist, from a metaphysical standpoint he's not wrong, either. However, another member of the Forsaken would contest the position- Demandred, who is said to have hated Rand's previous incarnation more than anyone has ever hated anything and has transferred that hate fully to Rand, making it plain that he sees himself as the only one allowed to defeat him though Moridin makes it plain he won't respect that if his plans call for it. Prior to his death, Sammael would have put himself up for the title too, though other Forsaken feel he falls well short of Demandred for sheer hatred- yeah, being the main character in a series where It Sucks to Be the Chosen One means you get lots of people who want this position. Other characters have them too, usually (but not always) their Evil Counterpart:
Mat is also considered the counterpart to the Forsaken Balthalmel (since both are gamblers and womanizers), though the two don't really interact. There is also Padain Fain, due to their mutual connection with Shadar Logoth, and Fain once held Mats family captive. And it is ultimately Mat who takes Fain down. Though Fain considers Rand his archenemy (despite hating Mat and Perrin also, adn the Ishamael and the Dark One), and furthermore Fain kills Perrin's family.
Redwall has had a few examples, but the most obvious are Swartt Sixclaw and Sunflash the Mace from Outcast of Redwall. When Sunflash maims Swartt's trademark sixclawed paw while escaping enslavement, a feud is struck that will haunt them both for their lives. From the time they are teenagers until Swartt's death in late middle age, they are always hunting one another.
Soviet spymaster Karla to George Smiley in various John le Carré novels.
In A Song of Ice and Fire, Cersei Lannister thinks that Tyrion is her worst enemy thanks to a prophecy stating that her she would be strangled to death by the hands of her younger brother. She is utterly convinced in A Feast for Crows and A Dance with Dragons that everything going wrong in her life is Tyrion's work.
More generally, House Stark considers House Lannister to be their archenemies and the conflict between the two is central to the story. The Lannisters see the Starks more as annoyances, but between killing Eddard Stark, holding Sansa and (supposedly) Arya Stark hostage, crippling Bran and masterminding the Red Wedding...Yeah, It's Personal. The Starks thought they were archenemies before, but by this stage several members of the Stark clan exist solely to destroy the Lannister House.
More specific examples, Cateyln despises Cersei and blames her for what happened to her husband and children; Arya seems to hate every single one of them but especially hates Joffrey; Joffrey thinks he is The Rival to Robb Stark (from a purely dynastic point of view, he is right, but Robb completely outclasses him as a warrior and commander, partly because Joffrey is still just a kid); and Robb himself probably saw Jaime as the Lannister he had to beat. On the flip side, the heads of these households, Tywin and Eddard, didn't seem to think about the other that much at all, while Tyrion Lannister has a...complicated relationship with all of them (both the Stark family and his own). The youngest children, on both sides, are treated more like victims of the whole thing.
Gregor Clegane, the Mountain that Rides, is an Ax-Crazysociopath with a penchant for Rape, Pillage, and Burn, and as such several characters, including his own brother, have sworn to kill him for whatever specific crime he happened to commit against them or their family on any given day.
Rhaegar Targaryen was highly regarded by most people and generally seen as The Wise Prince by everyone...except Robert Baratheon, who hated him with a such a passion that he not only killed Rhaegar personally in mortal combat, he sworn to kill every single Targaryen he could get his hands on, including their children. Though the practice of slaughtering the kin of rivals to the throne is not exactly uncommon in this setting (and the surviving Targayen children actually are plotting to overthrow him and his House), it is strongly implied that Robert did this purely out of spite for the Targaryens' in general and Rhaegar in particular.
In Timeline-191 the United States of America and the Confederate States of America are one another's archenemies, fighting four wars against one another, and spending most of their time when they are not at war preparing for the next one.
Cold Days, in The Dresden Files, reveals two factions have this kind of relationship: the Fae Winter Court against Nemesis, a Hate Plague/alien invasion. An epic-scale war that has continued for centuries.
Dime Novel hero Nick Carter's greatest foe was the fiendish Professor Jack Quartz, a hypnotist and vivisectionist.
The balor Errtu is the self-proclaimed archenemy of Drizzt. Errtu claims that no one else hates Drizzt as much as he does. Errtu is glad that Drizzt is a Dark Elf, a being whose projected lifespan can be measured in centuries, since this means there's a good chance Drizzt will still be alive when Errtu is eventually allowed to leave the Abyss. Every time Errtu finds a way out of the Abyss he makes a beeline for Drizzt. During one of his stays in the Abyss, Errtu took great pleasure in torturing the soul of one of Drizzt's friends. While Drizzt has other foes like the assassin Artemis Entreri and the orc king Obould Many-Arrows who both eventually make peace with Drizzt, Errtu is far more evil than either of those two and is also the only one who could theoretically hound Drizzt for the rest of his life.