@Junko7: You are basically naming my childhood! lol.
Final Fantasy games are some of my favorites, especially X-2.
Dark Cloud and Dark Cloud 2 are two I wish were on Steam! NEED those games back in my life.
I did play Radiata Stories too.
Pokemon Colosseum and Pokemon Channel, Ephemeral Fantasia, Harvest Moon games, Darkstone, Grandia 2, SSX, Dot Hack, Ratchet and Clank, and Jak and Daxter also give me a lot of nostalgia feels.

Oh, games that make me nostalgic.

There's a few but here's the ones I do play to this day, and I'm gonna give a quick overview/review:

Jade Cocoon [PS1]
One might argue that this is a Pokemon close, but it is so much more to me.
You play a silent protagonist who has to save his village from demons and does so by catching, training and combining monsters.
The overall style is way more mature than Pokemon and also deals with more mature issues. If you want to pick it up, I recon its hard to find but quite worth it. If you know what you're doing a playthrough of this game is about ~8 hours.

Dragon Quest Monsters/ Dragon Warrior Monsters [GBC]
Another pokemon clone, this time in the world of Dragon Quest .
The way you 'catch' monsters is by feeding them things to make them like you enough to want to stay with you.
You can breed monsters to create new species. This is a cruicial aspect of the game because with the monsters you find in the dungeons you wouldn't make it to the end. Always loved the style of the monsters ( it's dragon quest, of course the monster design is great) and the music really brings me back.

Lufia II: Rise of the Sinistrals [SNES]
This was for a while my favourite game to watch my older brother play.
It's a JRPG with a nice story, excellent spritework and don't get me started on the soundtrack.
If you want to play a oldschool JRPG that's not Final Fantasy or Dragon Quest, you should give it a shot! ^^