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I will always suggest Monster Hunter Stories, Spectrobes, and Fossil Fighters. There was also this monster catcher that had pretty decent monster designs, even if they looked like plush toys. Moka Moka I think? Or something like that.
There's also EvoCreo, and I recall KH Dream Drop Distance revolves around catching and raising Dream Eaters.
Though it's super tiny and I wish there were game based on it, Megaman Battle Network 3 lets you raise Viruses.
