The ones I know were already mentioned in this topic: the Harvest Moon series (I also recommend Rune Factory, by the same creators: it has the same gameplay but it takes place in a JRPG fantasy world with dungeons and monsters), the Chao Garden in Sonic Adventure, Black & White...
However, I agree with Ettina: the Creatures series is definitely the best "pet simulator" ever made (in fact, Ettins are a kind of creature from that series: I am guessing it is where Ettina's username comes from). You can get a free version (Docking Station, a standalone expansion of Creatures 3) in this link: http://www.creaturesdockingstation.com/ (I have not tested it, but you will need to use their offline patch, since the developer's server is currently down and some troll changed the password of the username they were sharing).
Additionally, the Monster Rancher series, available for PlayStation 1 and 2 and GameBoy Advance, is a very interesting variation of the Mon genre: you take care of a monster as if it were a pet, but you also train it to win monster battle tournaments at the Arena. And you can fuse your monsters to create stronger ones at the nearby Lab. Not to mention the monster creation system at the SHrines: using a password in the GBA games (any combination of characters will work) or a CD/DVD in the PS1/PS2 versions (again, any optical media you use will work).
_________________
DISCLAIMER: It should be noted that, while I strongly suspect I have Asperger's syndrome, I am not diagnosed. Nevertheless, my score on RAADS-R is 186, which makes me a pretty RAAD guy.
Sorry for this terrible joke, by the way.