10 pages long and only one person has mentioned Jade Cocoon: Story of the Tamamayu for the PS1? You should be ashamed. I have 2 copies of that game because I wore the first one out by playing it too much. The only way I can describe it is sorta anime + Dragon quest monsters + an RPG, but that really doesn't do the awesomeness of it justice. You can merge monsters together and create entirely new monsters, not like DQM where you just get a logical advancement.
Seriously, imagine playing pokemon, sending out your critter to fight, then just going 'actually my monster is a bit crap, lets just hit the enemy with a spear instead' and going and fighting the enemy with your weapons. It's plot and characters were awesome too, but it was a little hard to get into because of the strangeness(you capture new critters by blowing your flute at them, and you get money by spinning your monsters into silk. What. the. hell.)
It has a sequel, for the PS2, which sold terribly and was worse. 2 strayed from the fantastic original too much and made some strange new battle system and took out the merging and made your character a 10 year old boy with the mental age of a 3 year old. Also you couldn't even catch the critters, you had to hatch them from random eggs you found in the forest. It's barely even similar to the last one.
Also nthing Tombi!(or Tomba! if you're American), Future Cop LAPD(seriously everyone get out there and play that game now it's bloody fantastic), Populous: the beginning, Hogs of War, Rollcage and Creatures(I never really liked creatures, but damn it was well made)