Secret of Mana 2 is great (it's called Seiken Dentetsu 3 in Japan I think), though the combat feels a little different from the first it's basically exactly the same combat system. There's also an old SNES RPG called Robotrek, I think, which has you playing a young inventor boy in a modern world similar to earthbound. Your robots fight in battle, and instead of leveling up you design and upgrade them. It's a really good game that does a lot of things different but still feels polished. The enemies walk around on screen like in Chrono Trigger. Earthbound is also awesome. Haven't played that one in years. You pick your favorite food in the beginning, and some other stuff.
If you're willing to go a bit newer, you should definitely try the Grandia games. It's a great series (with no random encounters), and the third one has probably the best jrpg combat system ever (possibly better than Chrono Trigger's pure awesomeness). The only problem is that the story of the 3rd one goes downhill fast!
I never had a PS1 but I just played Vagrant Story recently and it's pretty good -- also recommended. There's also Legend of Mana, which can be as fun as the old Manas except it isn't tied together as well over a whole story.
If you're ever looking for a new RPG that feels like a really god 16 bit RPG, try Tales of Vesperia for the 360, it's a great game, and if you want it has decent multiplayer in battles. The enemies are visible in the field, like in Chrono Trigger (and most of the games I've named).
Hope you like whatever you end up trying,
Alic
