Although all the bosses in Shadow of the Colossus were basically beaten the same way (find the weakpoints, climb on the colossus and stab the weakpoints until it dies) most of them are INCREDIBLY unique. Special kudos goes to Phalanx, the amazing 13th colossus which flies high above the desert- the moment when you ride your horse alongside it as it draws low to the ground, leaping from the back of your mount to grab onto its wing and pull yourself onto its back as it soars high into the sky again, is worth an entire game's worth of awesome.
If you want an ORIGINAL boss, they don't get much more original than the battle with the Viola AI-possessed Ardjet from Zone of the Enders: the 2nd Runner. Because you can't harm Ardjet you have to time your attacks to parry its own until you get an opening to grab it and let your own frame's AI attempt to delete the Viola AI from Ardjet's systems.
The battles with Raziels corrupted brethren in Legacy of Kain: Soul Reaver (and his battle with Turel in Soul Reaver) are some of the best and most atmospheric examples of the puzzle boss I've ever played through, especially Dumah.
The boss battles from Legend of Zelda: Phantom Hourglass were absolutely awesome, especially the way some of them used the top screen. Most notable was the invisible boss which chases Link around the arena, and the only way to hit it is to aim via the boss' own first person view displayed on the top screen- when it charges Link you have to aim your bow until you can see Link aiming right into the camera and then let loose!
Aw hell, just read this list [http://tvtropes.org/pmwiki/pmwiki.php/CrowningMoment/VideoGameBosses] and pick out the original ones.
Angerwing said:
On-topic: The Bowser fight in Yoshi's Island was pretty unique.
Hell, most of the boss battles in Yoshi's Island were fairly unique. Remember Raphael the Raven? Being chased around the outside of a tiny moon (early shades of Mario Galaxy there) with 2 stakes pounded through it by a large black flightless bird, trying to beat it to the opposite side so you could drive one end of a stake into the surface of the moon, causing the other end to burst out the other side and hit the boss. It was AWESOME. Also see: the Potted Ghost, Prince Froggy and Navel Pirahna.