Mostly, it's the freedom. In many other video game RPGs, you end up being set in a certain character type, playing out a certain storyline. In Pokemon, you get your character, but that character isn't defined by dialogue or the people around them. They're essentially blank sheets that could have any sort of personality you give them. The storylines of the games are linear (well, in that there's no way to say, for example, that you run off and join the evil team instead), but how you finish it is completely different each time you play the game, depending on what you choose for your Pokemon team and how you go about fighting Team X, the Gym Leaders, and the Elite Four. In some cases, you can even earn your badges out of order, so it's easier to create your own story.
The media tends to be diverse as well. Unlike in many other fandoms, you don't see the same cast of characters and the same events in the various manga, the anime, and the games. The base (trainer sets off on a Pokemon journey) tends to be the same, but otherwise, the Pokemon franchise offers hundreds of different, diverse stories to tell its fans.
What this leads to is really a lot of freedom. I write a lot of fanfiction, and where I feel restricted to a certain cast or certain line of events in other fandoms (such as Harry Potter or Good Omens), Pokemon lets me be as creative as I want (without violating the extreme basics of the fandom -- like the six Pokemon rule) because there's no universal canon. It even offers ultimate freedom in diverse character possibilities (breeder, coordinator, researcher) so that I don't even have to write about trainers if I don't want to.
So, really, the openness of the fandom is what keeps me here. The games and other media tend to be addictive too, but my interest in those fades in and out, whereas the fanfiction possibility is really always there.