Will there be unranked and ranked player vs player battles or even tournaments?
If you win a tournament (ranked or not), will you get acknowledgement from it? Like will there be a global ranking system so you can strive to actually become the "very best like no one ever was".
Hold a title. Like if you keep what they used to do when you pushed A on a tv, and it shows your latest accomplishments. Instead it could perhaps hold information dealing with winners of tournaments, number 1 ranked local/regional/globally rated player: who has the most caught Pokemon, who has the most wins against other players; and/or sightings of rare Pokemon (I read that you have plans so I really thought about not typing that but if it was set in stone, I thought it might be a nice thing to consider. Just a thought.)
I know that not everybody likes the whole someone's "better" than me because they've battled more or anything, but being able to say, "I AM the very best" would be such an accomplishment. there's fame, and always people after your title, it won't get boring too soon. (All of this is obviously from my point of view. I have no idea how other people feel about this topic.)

Awesome, and thank you all for your enthusiasm in helping people, haha.