For boss battles, we're talking Devil May Cry.
Angelo Credo was a great boss fight. You could really play around with him, because his attack patterns allowed for creative ways of fighting him. My favourite was to get him to charge at me with his Aegis Shield by using the Buster's Snatch ability, and then try to Buster-grab him, causing him to fire off a two-hit combo, but leaving him with an opening for a Buster-throw. Gooood times. Hell, the Bianco Angelos were good fun, too. Like miniature versions of Credo.
But if you want a REAL boss fight, and not someone to just muck around with, you want (depending on which game you're playing) Vergil or Dante.
In Devil May Cry 3, you fought Vergil 3 times. And each time, he was a more entertaining opponent. During the first fight I had with him, I was using Trickster, which was an absolutely perfect match for his Dark Slayer style. With each swing of his Katana, I was able to dodge or leap out of the way, just in time... until he started copying me, dashing around the arena along side me, not even letting me stand still for an instant. For every dash I took, he took a bigger one, and added in an iaijutsu strike (quick-draw katana style)! It was an awesome battle, bookended by some amazingly choreographed cinematic fights. Seeing Vergil calmly collect and return Dante's bullets was awesome. The second time you got to face Vergil, he'd picked up a new set of toys - the Beowulf armour. With these gauntlets and boots, he was an unstoppble close combat machine - perfect for my Swordmaster. For each of his brawling moves, I could counter with a flurry of sword swings from Rebellion, or use a tornado from Agni and Rudra to push him back just enough to catch my breath. It got better when he started to use Beowulf in combination with Yamato; the attacks kept coming. And then he used his Devil Trigger! Nothing quite as frightening as being backed into a corner by a barrage of fists and sword swings, and then, when I try to counter with my ultimate transformation, HE DOES IT TOO. Amazing experience. But the third fight, the hardest, was the best. See, Vergil picked up Force Edge, the broadsword wielded by Sparda, Father of Vergil and Dante. So now, not only does he have Beowulf and Yamato, he's got a copy of my Rebellion. So when I try to Stinger, he just dashes and does it back, and when I try to do an uppercut into a helmet breaker downward slash, he just mirrors my moves. Damn near unstoppable, it took me a good half hour - on NORMAL - to beat this guy. To be fair, I was playing on PC, which meant gimped controls, but still... he was an impressive force. But DMC4 had an even better boss.
They absolutely nailed the AI in Devil May Cry 4 for Dante when you fight him as Nero (the second time, of course). So when they introduced a new guy, I was all skeptical. He'd never be as fun to play as Dante. Nero, this guy who looks very similar (the difference is limited to a blue/red overcoat instead of just crimson, and a glowey blue arm), doesn't even have two guns, and has a gimmicky (or so I thought) sword... how could he ever match up to the Legendary Demon Hunter, Dante? Well, for one, he doesn't need two guns, because his gun has two barrels. It doesn't fire a constant stream, but it fires bursts, which work almost as well. And while Nero lacks the specialised combat styles of Dante, he's got his Demonic Arm, which adds a whole new dimension to fighting. Instead of just slash-slash-slash, Nero is all about grab-slash-throw. With the arm, you can snatch people from across the room (sort of like Scorpion in Mortal Combat, except without a rope, and without a spike on the end). So, this sounds cool, but against a boss, wouldn't it be overpowered? Well, most bosses resist a Snatch attempt. Except Dante. Who kicked your ass if you Snatched at him at a bad time. And dodged it only most of the time.
The great thing about Dante is how closely he mirrors the way you fight. Sort of like Vergil in DMC3, but even more similarly, because many moves are literally shared between Nero and Dante. Even the fact that we both have guns. Whenever you fire the Blue Rose, which is Nero's double-pistol, at Dante, he'll counter with the same number of bullets from Ebony and Ivory (his pistols). So on any other boss or enemy, after leaping away, or whatever, you'd let loose a barrage of fire from good ol' Rose, and keep the enemy on the back foot. Doesn't work on Dante. So, how about swords? Well, Dante's Rebellion and Nero's Red Queen are bloodly similar in terms of attacks. There's a twist (no pun intended) with Red Queen, but I'll get to that in a moment. So, Nero/Dante each have a standard pool of moves - charge, uppercut, downward slice, 3-4 move combo, aerial combo, etc. As expected of Dante, he's very hard to actually hit with a sword swing. He'll duck and roll and jump (and dash, if he's in Trickster) away from most attacks. Until he gets bored of running, and begins to mirror attacks for the hell of it, causing both of us to be stunned for a moment. And then, because he can, he begins to perform attacks quicker. So not only can he do what I can, he can do it BETTER. Ah, but Red Queen has a twist. The handle of the blade can be revved like a motorcycle's handlebar, causing a fuel to spread across the blade and be ignited when you swing. As such, Red Queen bursts into flame. A lot. It's more likely to hit Dante when it's on fire, and the awesome factor goes through the roof.
Remember how I said Nero had a demonic arm? Well, against most bosses, there's a moment where they're stunned or open, or whatever, and you can grab them then to perform a cinematic beating, and deal a massive amount of damage (sort of the inverse of what's standard of a boss fight, where they lock you down and combo you to death). Well, against Dante, if you try the Buster, most of the time, he'll COUNTER, and force you to button mash in a strength contest with him. And if you fail, then YOU'RE stunned, not him. And he'll proceed to cut you apart.
I was extremely impressed when I and he would perform mirrored attack sections - I'd charge, he'd roll, he'd charge, I'd jump into a slicing meteor strike, he'd jump and do the same! The most amazing feeling comes from fighting a boss who's just as good as you are, and you can see it. And then he kicks it up a notch, and begins to switch styles. Even as an enemy, he's got all four of his combat styles, and each of them is deadly. From countering my aerial combo with a Fireworks display from his shotgun to using his Royal Counter to knock me away to using his swordmaster chain combos to cut me apart... I was defeated repeatedly, but I was loving it. Dante in DMC4 is the epitome of a boss battle, in my books.
...that's a lot of words, in retrospect. Sorry, for people who expected a nice light threat about fun bosses. I had to get all lengthy, didn't I?