I know I'm not digging as deep as a lot of other people on this one, but Buffy's archnemesis (archnemesis's?) were the yearly big bads. The difference between her and comic book heroes (and despite what others have said, Buffy is totally a superhero) is that Buffy's "big bads" tend to get killed.
I mean, when you think about it, The Master was the complete focus of Buffy's Slaying attention in S1, but he died in the end. Angelus was absolutely the arch-enemy of Buffy in S2, but he gets transformed back into Angel at the end of becoming. Same story for The Mayor. Faith is probably the closest Buffy gets to a true, classic "arch-nemesis", mostly because they're rivals for so long, but in the end they reunite. Adam is a weak villain, but like many of the others he gets seriously dead. Glory is a huge arch-enemy and probably would have remained so had Giles not killed Ben. The First in S7 fills that role as well.
I think at various times, all of those were arch-enemies of Buffy. In comics, which unlike T.V. are ongoing, they can't just kill off the top and most interesting villains every year because they'd run out. It's different with television. Sure, Angel had Wolfram & Hart, but they were a heavy metaphor and more of an institution (as opposed to being singular villains like The Joker or Lex Luthor are). Angel always had bad guys, even big ones (thinking of guys like The Beast), but Wolfram & Hart was always looming over everything.
I don't think slaying was Buffy's arch-nemesis. I really don't, no more than Angel's soul was his. Slaying was a responsibility that was thrust upon her, sure, but she chose to continue doing it, just as Angel chose to fight the good fight. That was an incredibly heroic decision, and at the end of the day, that's just what it was: a decision. She and Angel chose to keep fighting. Any day they could have stopped, but they never did (I think that was best exemplified in "I will remember you"), because that's what heroes do. I don't think that's her "nemesis" at all. It brings out one of her biggest strengths (God, when I get going about Buffy...I just love her so much...🙂)