The Diablo 2 battles were always hard for me, because my computer at the time was slow. It basically went: walk up to boss, lag, dead. Not fun, although I did like the game as a whole.
Now for my favorite boss battles:
Raven (your character in Armored Core) vs. Ares
Akuma vs. Gen-This was the boss battle for Akuma in SFA 2. For some reason, I always had trouble with it.
Jack vs. Kyorta (Robotech: Battlecry)-two aces going head-to-head. Gotta love it.
The final boss is boring by comparison.
Falcon vs. Valgas-Power Stone
Gan Ning (or anyone) vs. Lu Bu-not really a boss, but he is a phenom in one of the levels that you can fight if you want a challenge. Gan Ning is just my favorite Dynasty Warriors general.
Mitsurugi, Nightmare, or Astaroth vs. Inferno (Soul Calibur 1/2)
And you can't forget the most important one:
Me vs. my friends (any game)
I suppose the overarching theme of all of these is that they are 1-on-1 battles where you are in complete control over your character. I like the whole "duel" scenario.