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Honestly I prefer a deeper story that makes me think about the implications of what I'm doing and asks questions about the worldbuilding involved, thus helping cement the game as a real and living world instead of just a power fantasy autism grind simulator. BW was probably the closest I'll ever get to that, even if it didn't quite clear the bar because it entirely hinges on N not really knowing how Pokemon work, and the player character being objectively correct. If I just wanted to see numbers go up and down I'd go play on Showdown or play a Battle Tower, not buy a whole new game. I'm here to experience what the game itself can supply on its own and can't just be copied by scrolling up and down a google spreadsheet full of bigger numbers.
Mechanically the games don't have to do anything special, although I love it when a game also has deeper strategies that Pokemon regularly misses. Colosseum and Gale of Darkness probably got the closest there, but again they kind of just don't entirely clear the bar, but that's not really the point of the thread.
It doesn't help both of my preferences that Pokemon has just kind of tossed both out the window at this point by doing low stakes stories that work regardless of worldbuilding (Arven and Penny) or just aren't explained properly and thus don't actually matter to the world despite having the appearance of mattering to the world (individual Paradox Pokemon just don't make sense to me, Convergent Species don't add up as needing to be something special when we've had much more comparable analogues for ages, Regional Forms/Evolution just bother me now because how come one region gets objectively better versions of the Pokemon?). I don't even hate the idea of low-stakes stories but when it feels like the kind of thing I could get basically any story, why would I bother going to Pokemon specifically? A story should inform me at least a little of the world, and the world should help shape the stories in my opinion.