Armored Core
Ah yes, another good one!
I bet Armored Core, has not aged well.
The original hasn't, unfortunately. But it's still a good game, if you don't mind the blocky graphics. Seconding it regardless, for nostalgia.
Ehrgeiz is a good one too. Has a couple games in one, but I spent entirely too much time on the dungeon portion (I was never really a fighting game guy). Has cameos of Cloud, Tifa, Yuffie and Sephiroth in the fighter portions, though.
Front Mission may be worth checking out... the original in the series is one of the better ones, but it's not for everyone.
Gex. <3
Second on Jade Cocoon. First one is better, imo, as the second one seemed to get dumbed down while increasing overall playability (but the second one is on PS2, so that's to be expected).
LSD is a strange one. I found it on a dream site (or rather, a link to it) and it is just... well, play it and find out. I can't spoil the surprise because every time is different. Japan only, so you need an emulator.
Mega man series... kinda a given, considering a good portion of it was on the PS1 and it didn't seem to lose its NES/SNES feel as much.
Monster Rancher; another Pokemon-ish game with some interesting features.
Populus: The Beginning takes the series in a slightly different direction... not unpleasant, but different. Had some fun with it for a while regardless.
Rampage series: Probably one of my favorite time-wasters from when I was a kid. It's what got me into the whole "Fully destructible? No problem!" mindset.
Finally, Romance of the Three Kingdoms series is one of the best strategy games on the platform. I loved these, considering I didn't always have a computer at my disposal and they translated to controllers well. Up until the most recent entry, anyway...
Edit: Almost forgot about Theme Hospital. Priceless game, one of Bullfrog's last.