legend of dragoon
This was the first game I played where I actually planned out how I was going to fight before hand. I'd do a test battle against the boss to see how far I could get, what they could do,to get a rough idea of their total hit points, and then I'd get slaughtered and start planning my strategy. I'd figure out how much damage my team could do in a turn, and I'd figure out how many turns it would take to kill the boss. There were many times when if I screwed up during the "hey, it is a good idea to play Simon in the middle of our battle engine, ha ha, have fun" button sequence for one character I would just reset and start over because I knew I was off rhythm and I'd end up wasting turns healing, thus making the battle un-winnable. The Divine dragon fight was really good. Melbu Frahma was just annoying as hell though.
One of my most memorable boss fights has to be Kain R. Heinlein from Garou:Mark of the Wolves. Using Terry, I got to him on the highest difficulty setting. In the first match I was flawless against him, I simply lucked into every move I needed to beat him to the punch every single time. The second round, I don't think I hit him once, or even came close. A friend of mine was watching and said "You pissed him off now, watch out." The last round was really close, like something out of a movie. We were both at "one hit will kill you" health when I landed the final attack.
The final boss(es) of the first Act Raiser also comes to mind.