So what would your game be?
Think... World of Warcraft: Pokemon.
Its an MMO, you can play a trainer if you wish, but you aren't necessarily required to. You could be a breeder, or a farmer, or a "professor", or a team rocket member, or etc etc.
You can have a maximum of 6 pokemon in your party at one time and you can battle trainers or wilds as you see fit. The battles can be automated with the regular pokemon battle system, but pokemon in this can have more than merely 4 abilities. Or you could go into like.. an FPS mode during battle and actively control your pokemon, skills by skill, or just make them trade blows.
There would be quests like any MMO would have. The game would be constantly updated with new pokemon that you could battle and capture if you had enough skill to do so. Rattatta would be significantly easier to capture than a Rapidash, for instance. Legendaries would be unobtainable by anyone, or you could obtain them for a quest but could not use in the "real world" of the game.
You would be able to use your pokemon as mounts, any one that would make sense to be able to do so with. You could accessorize your pokemon much like you can with the fashion case. Your pokemon would no longer evolve, but as they leveled up would sort of slowly start morphing into their other forms (this may not work too well I suppose as some pkmn are just too different to have that go well) Stone evolution would work the same way, they have to hold the stone and would, more rapidly than normal, evo over some time. Zoroark and Ditto would be very interesting ones to hunt down, yes?
I havent thought up an interesting crafting or "farming" system yet, but I do intend to add something like that to this... Pokeball crafting or something. A new ball type for every pokemon type and some more interesting ones like a Sun Ball, opposite the dusk ball.
General ideas, wishes, and more...
What platform would it be made on?
Obviously it'd have to be Nintendo but in a perfect world it'd be on every console and PC AND be a cross platform game. PC, XBox, Wii, and PSN players all play in the same servers.
What would the story be?
Open story, some main plots to choose from and many small side stories, always changing as the game is updated and player influence changes the world around them. Players could make their own quests for others to perform as well.
Who would be the hero/villain, if there is one at all.
the Team Rocket style people wouldnt necessarily be the villains but they would be players/npcs to keep an eye on.
Would there be more than one game?
I don't know. WoW and City of Heroes seem to do just fine without making a sequel every other month so I suppose not, just lots of updates.