Ice Titan
Fought in the Olympia Stadium, the Ice Titan isn't as hard a fight as it is long. His attacks follow in a pattern that is pretty consistent. Fighting him may take some time, but it isn't a total challenge.
Phantom
Fought in Neverland, if we come back there near the end of the game. We must go to a room with a save point and a pixie, then take her issue. Peter must be in our team, because only he can understand the pixie. Battle is hard for two reasons: we cannot use summons (what means no auto-life) and the Phantom uses curse that kills instantly when the countdown reaches zero. To stop the countdown and buy some time we must cast any of stop spells near the clocktower.
Kurt Zisa
Fought in Agrabah, if we come back there near the end of the game. We must go to the Alladin's room and take issue from the Flying Carpet.
Sephiroth
Fought in Olympia Stadium in the Platinum Cup. Sephiroth is seen as the hardest boss in Kingdom Hearts ( original version, not Final Mix). After his HP meter reaches pink, he uses Sin Harvest attack, which reduces HP to 1 and MP to 0 and rush on you with a following attack. If he teleports close to you, you can cancel the attack by hitting him. If he is too far from you - run as far as possible and use an elixir (you have very little time to do it). After his HP meter reaches orange, he starts using an annoing chain of flying attacks. After his HP meter reaches green, he starts using Meteor (you cannot damage him during the animation) and attacks you with orbs of dark energy.
Enigmatic Man (Final Mix only)
The hardest boss in Final Mix. He attacks in the same way as Xemnas in Kingdom Hearts II and is found in the room in Hollow Bastion where you fought Maleficent Dragon.