PS2
Radiata Stories - Quirky and fun ARPG with two different story-lines to follow.
The Bouncer - Very early PS2 title from SquareEnix. Unique little beat-em up combat system; events in the story are seen through three different playable characters.
Shadow of the Colossus - One the most highly regarded games of all time. You're a boy killing giants to save the girl you love.
Okami - It's kinda like Zelda, but you're a goddess dog who uses a paint brush in battle.
Stella Deus: The Gate of Eternity - A decent little tactical, turn-based RPG. Has some outrageous attacks, and plenty of recruitable characters.
Front Mission 4 - Another tactical, turn based RPG, but this time with giant robots. I personally love the game, and you don't need to play the prior games in the series.
Romance of the Three Kingdoms - If you like strategy games, try any game from VII to XI. You build up cities, fight battles, have duels, and engage in debates.
Metal Gear Solid 2 - Kinda polarizing, but I personally enjoy this very very much.
Soul Calibur III - You have the second one, so try the third one. It has a pretty neat board-game style campaign, with a story focused on your created character.
Final Fantasy XII - People give this one shit, but I remember liking it. Decent combat system that allows you to experiment with AI behavior.
Rogue Galaxy - A very anime ARPG, but a fun game with a variety of different locations.
Mobile Suit Gundam: Federation vs Zeon - Pilot giant robots, and fight other giant robots.
Mobile Suit Gundam: Encounters in Space - Pilot giant robots, and fight other giant robots... in space! A much faster game than the other one listed.
Mobile Suit Gundam: Zeonic Front - Kinda clunky, but one of the finest Gundam games, and a very meticulously tactical game. Plan your assault by dictating where and when your AI allies will strike.
Def Jam: Fight for NY - Pretty fun fighting game with a hip-hop theme, and a story mode.
PSone - You'll probably have to hit Amazon or Ebay for some of these.
Brave Fencer Musashi - A very very strong ARPG with a world that felt amazingly alive for a PSone game.
Saga Frontier II - An RPG with a story that spans decades and more than one protagonist. Can be very challenging.
Final Fantasy Tactics - The title tells you what it is. One of my favorite games of all time.
Legend of Legaia - Very unique RPG battle system, where you chain marital arts moves together to create new combos and techniques.