There are too many to chose from...
1) Final Fantasy 7. Enjoying playing this at the moment, although finding it a little too easy... Might've gone through too slowly!
2) Gran Turismo/3/4. Despite the fact that as kids I never got the IA licence on the original... but we loooooved racing splitscreen - MX5 vs. FTO was the main event! Still have the occasional blat on GT4 with my Elise on the Nurburgring.
That's it, since the third spot is going to have to be shared by:
Metal Gear Solid - amazing single player experience.
Perfect Dark - so much fun playing splitscreen with masses of variety.
Shadow of the Colossus - just a perfect game despite only going through it once. Dripping with atmosphere.
Wipeout 2097 - Shouldn't need to justify this...
Timesplitters 2 - The first one was great in it's own way but this one took it to another level

Only gripe was the minigun wasn't as meaty as the first... Humour, fast paced action and ducks with tommy guns.