Yeah, I really wish IVs never existed.
Anyway, you kind of raise monsters/dragons in Bahamut Lagoon, and it's a pretty awesome game, but it's not exactly the kind of monster raising you're talking about. It's a strategy RPG, and the dragons are more of a side project for helping you in battle. The dragons also come to you at set times, like after you complete a certain mission.
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Essy, Avian Enthusiast