Welcome, Emily. ^^ I'm a lurker most of time since I'm currently unable to play games much (and until a few months ago I had no PS2 >< missing all of the "new" generation after the original PS/Saturn/N64) :P
But anyways, I like to replay most games I finish, but sometimes I dont get to finish the second run. The ones I did replay and finished are:
Phantasy Star IV: Love it up to this day. The first electronic RPG I had the pleasure of trying out \o\~ I never had it myself, just rented about every weekend for months during middle school XD And I never had the bad luck of having someone else renting it in the middle of the week and erasing/overwritting my File. But yeah, I just had a lot of fun playing it and replayed about 7 times. There was a point where I played from the start and got to finish it before the weekend ended x.x
Final Fantasy IV: Ok, theorically I didn't "replay" since I only got to finish it once. Basically I played it on SNES, reached the last boss, but couldnt beat it. After many attempts I got fed up and quit trying. Many years later I get the PSX remake and try the game again, finally reaching the last boss once more and killing it after over 30 (no joke) attempts on the last boss.
Final Fantasy VII: For a long time it was my only game on PSX. As a result I played it and replayed several times. Even used GameShark at a point ot get access to Sephiroth, as well as Aeris. It was fun beating the Weapons with her. Sephiroth sucked because I never managed to find out the code to control him, if there was one, so he kept on casting random elemental shit, which cured Emerald >< It's a fun game, and I might try it again soon. Need to try some odd MATERIA combos as well as rewatch the storyline.
Final Fantasy VIII: My second PSX game. I was already kinda burnt out of playing FFVII and beating it in every way possible. I loved FF8, in fact it's my second favorite FF up to this day. Specially because of the setting on which the story takes place. Who never dreamed of studying in a Garden? X) And I plan to try that again soon, because I never got to make a "perfect file" on it despite replaying about 2-3 times. Quistis rules.
Final Fantasy Tactics: My favorite FF. I never got to make a perfect file on it either, and I never owned it (rented and later on borrowed from a friend). Anyone wanna send me a copy? A PSP with the upcoming remake also works lol >< The storyline is amazing, and it was what got me replaying multiple times, though the battle system and character customization helped on that X) I also used to know the logic behind some GameShark Codes back then, and with that I was able to create my own jobs and stuff like that on my 3rd replay. It could get 'buggy' at random points though :/
Xenogears: One of the best RPGs ever. I've replayed this one 2 times, but oh wow, they were totally worth it. After the second replay, I just worked on perfecting my file. Sometimes when bored I would just turn it on and do some deathblows on random mobs to pass the time. As a result my characters were all lvl 85+ when I stopped playing this game XD The Kislev Arena was pretty fun too, I liked playing versus my friends there 8D~ As good as some fighting games.
Chrono Trigger: I find its storyline pretty weak. But the OST, along with the game system and the character design really draw me in. I played it a lot just to kill random mobs, like Xenogears. Got all endings. Oh, I did it on the PSX remake, never played on SNES (my cousin said I was "too dumb to understand RPGs" when I was little and saw it for the first time, it was a year before I played PSIV, my first RPG as mentioned above).
Chrono Cross: I like it more than CT. Finished it the first time, then before I started a New Game+, my Memory Card was damaged and I lost all data in it T_T I ended up replaying it, finished again. Started, but never got to finish a New Game+ though.
Legend of Mana: Not sure why I replayed this. But I did. It was fun, lots of customization, the several plots are light and some characters just have charisma (Elazul and Pearl's plot was my favorite).
These are the ones I remember, there may be more I've replayed but dont come up to my mind right now X)