I want to say Gamecube because since it has been so under-appreciated and I got it for such a great price... but overall I'd have to go with PS2. Even though my first one died on me and I had to buy a slimline to replace it, one reason I still use it is the excellent backwards compatibility with PS1 games plus how it totally dominated so many franchises in terms of exclusives.
Games that come to mind:
-MGS3
-Devil May cry
-GTA 3, VC and SA
-Tomb Raider Anniversary and Legends
-Red Dead Revolver
-Black
-Blood Omen 2
-Resident Evil Code Veronica
-RE4
-Hitman 1-4
-Silent Hill 2 and 3
Yet at the same time I realise only 3 of those games are PS2 exclusives
Gamecube had all of these EXCLUSIVES:
-Metroid Prime 1 + 2
-Zelda Twilight Princess (well... exclusive for its generation)
-Zelda: Wind Waker
-Eternal Darkness
-Resident Evil (remake) + RE: Zero
-Rogue Squadron
-F-zero GX
On top of that Gamecube usually won out in terms of quality of graphics in multiplatform games thanks to its advanced ATI designed processor (who would later design the hardware for the Xbox 360) that transcended any limitation from mini-DVD just like 360's DVD trounced Blu-ray.
On reflection, I'd say it is a dead heat but for balance I'll give the underdog Gamecube the award, it was the last solid home console from Nintendo with a roster of games to be proud of, shame it just couldn't shake off its "kids console" image.