I think Final Fantasy X-2 is the FF, where the International version gives you the most extras. I bought a flip top just for that game - and I can say it was worth it.

Extras:

-2 new DS: Psychiccer and Festivalgoer
-You can capture fiends, monsters and even NPCs and put them into your party.
-You may fight in a colosseum, where you find monster cups and the special Farplane Cup...
-Using the colosseum you can bring four fiends into your party, which you normally wouldn't have gotten.
-Fightable in Youth League Cup: Brother + Buddy, Yaibal + Elma + Lucille, Nooj + Gippal + Baralai, Lulu + Kimahri, Three Leblanc Goons.
-.....


Farplane CupContains Tidus, Auron, Seymor, Trema, Concerher, Paragon, Azi Dahaka, Elder Drake, Black Elemental + Mushroom, Insect Matriarch and the worst: Major Numerus