Dr. Horrible's Sing-Along Blog - Johnny Snow claims to be Dr. Horrible's nemesis. Similar to the Buffy example above, Dr. Horrible does not take him very seriously, considering Captain Hammer to be his actual nemesis. For his part, Captain Hammer thinks of Dr. Horrible as just another villain that he regularly defeats, and a fairly pathetic one at that.
Dr. Horrible "OK. Dude. You're not my nemesis. My nemesis is Captain Hammer. Captain Hammer, corporate tool! He dislocated my shoulder last week. Again."
Atop the Fourth Wall and The Spoony Experiment - Dr. Insano was both Spoony One's and Linkara's nemesis. Lately, though, Linkara's new archnemesis has become Lord Vycenote because getting Spoony to appear in his videos all the time is logistically unfeasible, and Insano is now more of an annoyance to Spoony.
Mechakara is a better example, as he is Linkara's most determined enemy. He has been destroyed twice and came back stronger then ever.
In Survival of the Fittest, both the winner of V1 and the winner of V2 have their own arch enemies. The V1 Winner, Adam Dodd, has Cody Jenson, who killed his girlfriend, raped one of his friends, and beat his other friend into unconsciousness. The V2 Winner, Bryan Calvert, has Mariavel Varella, who killed his girlfriend after swearing vengeance on Bryan for killing her boyfriend. Neil Sinclair (who was not the winner, but definitely the hero) of V3 had Danya himself.
Agamemnon Tiberius Vacuum claims Brooke Brodack is his nemesis (a claim she embraces). What exactly the past relationship between the two that lead to this is, is still unclear.
In the Hitler Rants parody, Hitler's archnemesis is resident trickster Hermann Fegelein... the former is always at the butt end of the rivalry.
In the 2nd sequel, Ringman actually becomes Tiamat's arch enemy, even though he personally never laid a glove on her; then because of his bid to free the centaurs, Ringman even becomes the Dungeon Master's arch enemy.
In his MinecraftLet's Plays, the Evil Chicken is this to Gronkh. It's uncertain whether he considers Evil Chicken a specific creature which keeps reappearing to ambush him, or the chickenhood is not only evil but his personal arch enemy.
Spikerbleeper - Arch Enemy of Mr. TARDIS.
[[STAR]] has a huge hatred for sentries, but what he hates the most is the Mini-Sentry from Team Fortress 2. Justified since it's the no. 1 thing that ruins EVERYTHING for him. The worst thing that happened to him was playing the perfect loadout to get ruined by a Mini-Sentry.
Rythian considers Sjin and Duncan Jones to be this throughout The Blackrock Chronicles, since they destroyed the old world and routinely harass him. They eventually reconcile.
RWBY: Volume 2 seems to be setting up this dynamic with the heroines and the villains. Ruby has Cinder, Blake has Roman, Yang has Neo and Weiss has the White Fang in general. Especially Blake and Roman.
The Executive from Fallout Is Dragons fits this trope for Powder Keg, as he forced the young unicorn to shoot his own brother for no reason other than his own amusement.