It's too bad you're getting a PS2 now, seeing as several good games will be hard to get without going broke, as it's been mentioned.
As far as non-Square-Enix RPG's go, try out Shadow Hearts: Covenant and From the New World, the PS2 Wild ARMs titles, Persona 4 (P3FES is also recommendable, but it may be unplayable for you if you play P4 before it), Grandia III, Tales of the Abyss, Breath of Fire V: Dragon Quarter, Dragon Quest VIII: Journey of the Cursed King, the .hack//G.U. games, the Xenosaga series (except Episode II)... well, there's a lot. On the more challenging (read: unforgiving, may make you suffer) spectrum, there's the ever-recommendable Shin Megami Tensei III: Nocturne and both Digital Devil Saga titles, but as it's been mentioned, they'll be hard to get at a sensible price. Bargain bins are always your friends.
And while I'm sucking up to ATLUS I'll also mention that Devil Summoner 2: Raidou Kuzunoha vs. King Abaddon is coming out soon in NA, and it's a great game. But you'll have to play the first Devil Summoner to fully understand the plot, and that one's not so good. Might also be expensive.
In other genres I recommend the Fatal Frame trilogy and Rule of Rose (survival horror; RoR is a bit of an acquired taste), Ratchet & Clank: Up Your Arsenal (action shooting), Rez and Space Channel 5: Special Edition (rhythm), Devil May Cry 3: Special Edition (action), Tokobot Plus (platforming adventure), Robot Alchemic Drive (mecha action), among many others, and possibly my favorite PS2 game, Ico.
Let's just say that the PS2 has tons of great games and you should feel bad for getting one this late.