Being a Nintendo fan, the only PS console I owned before is the PSP (aka my Monster Hunter machine). I did already have some PS2 games in my mind to get (Xenosaga, Star Ocean, ICO), but it's always nice to have other opinions!

EDIT: Thank you so much everyone! I never expected that many replies XD

After looking through the replies, here's what I got (after weeding out the games that I am not that interested):

Xenosaga I/II/III
Xenogears
Star Ocean
Metal Gears series
Dark Cloud 2
Rogue Galaxy
Shadow of the Colossus
ICO
Bully
Persona 3
Persona 4
Disgaea 1/2
Resident Evil 4
Final Fantasy X/XII
Devil May Cry 1/3
Onimusha series
God of War I/II
Silent Hill 2
Haunting Ground
Rule of Rose
Forbidden Siren
Forbidden Siren 2
Kessen 1-3
Timesplitters 1-3
Kingdom Hearts 1 and 2
Kingdom Hearts: Chain of Memories
Jak and Daxter series
Sly Cooper series
Gran Turismo 3/4
Ace Combat 4/5/0
Katamari Damacy
Dragon Quest VIII
Shadow Hearts Covenant
Prince of Persia
Suikoden 3
Megaman X Collection
Onimusha
Wild Arms
Mana series
Ar tonelico
Grandia 2 & 3
Castlevania: Symphony of the Night
God Hand
Yakuza 1&2
the SMTs
Viewtiful Joe
Legacy of Kain series

I might get some of them for different systems instead of the PS2 version, but holy shit that's impressive... Being interested in most genres doesn't really help either.

You can play every canon entry in the Metal Gear series up to MGS3. Go for it. Remember, you have most, if not the entirety of the PS1 library to work with, so feel free to go nuts.

Silent Hill 2

Haunting Ground, Rule of Rose, both Forbidden Siren games, (maybe) Obscure and of course, Metal Gear Solid 2: Sons of Liberty.

It's certainly not fixed for me, that's for sure.

Kingdom Hearts 1 and 2. Re:Chain of Memories(if you can get through the card style gameplay, which has mixed reactions).

Zone of the Enders 1 and 2.

Jak series.

Sly Cooper series.

PERSONA 4

If i have a ps2 id buy yakuza 1 and 2