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These are the most basic things that keep a monster game going. I'm stripping things down to it's simple core, so things like customization and day/night cycles aren't listed.
Please let me know if I'm missing anything.
>1: Good designs
Pokemon has these which is why it stays on top despite everything else slipping. Monster Rancher and YoKai Watch are close contenders.
>2: Balanced Gameplay
Pokemon doesn't have this outside of the Mystery Dungeon spinoff series. Period. Monster Rancher was closer with most monsters of a certain rank being similar in power level. If memory serves Digimon had/has decent balance but crap designs.
>3: Good Lore/Worldbuilding
Pokemon got this right early on which killed off Monster Rancher. Monster Rancher, despite being a better game didn't have much lore and what was there was gloomy. The anime was a mess that explained itself in the final episode which was actually banned in the USA for being depressing. The Worldbuilding on Digimon made it survive having meh monster designs but the theme of "computer monsters" was too 90's to last once the dot com bubble burst.
Mix good designs, balanced gameplay and good worldbuilding and you have a winner.