The main things that I thought of that might not be on every mind are the music, RNG, and glitches.
The music from the games was all I ever listened to for a very, very long time, and what inspired me to start writing my own (even in the stuff I write nowadays I occasionally sneak in a soundfont ripped from one of the GBA or DS installments.) The soundtracks to Ruby & Sapphire, Card GB 2, and Black & White 2 are especially precious to me, and there are plenty of great tracks amongst the rest that have helped me preserve countless powerful memories.
As for RNG, while it wasn't something intended to become a focal point, it became the main reason I kept playing the games around the BW days. I loved applying the skills involved, starting out with breeding in SoulSilver and gradually expanding the repertoire. RNG in Gen III became the most enjoyable one to me, due to the utility of emulators to cut the corners time-wise without ever directly altering the Pokémon data, the potential challenge of obtaining the same Pokémon on a retail cart (or using stuff obtained on an emulator on a physical copy!), as well as my general love for the Hoenn games. Scrolling through seeds in the Researcher on RNG Reporter may have been intended as a means to an end for players and a complete externality for the game developers, but to me it was legitimately enjoyable. I kept a trade thread where I listed the stuff I got, but I almost never went through with trades or responded to CMTs because I didn't care deep down about building a huge library of competitively viable monsters, I just liked the process of obtaining and admiring them on my own copies of the games.
Glitches are the most minor of the three things here, but my interest in them goes all the way back to elementary school where I'd hear rumors about MissingNo. and foam at the mouth at the thought of testing them out on my friend's Red. I ended up experimenting quite a lot with glitches over time (though hardly at all outside RBY), and while I never made any major contributions to the ongoing studies, observing and playing with the incredible amount of exploitability in the coding loopholes that emerged over the six years of the original Kanto's development remains an activity of incredible interest to me. Amidst the devious and benign Old Man and Escape glitches, the sheer destructive force of the ZZAZZ glitch and Cooltrainer move, the aesthetic appeal of Glitch City, the delicate process of capturing glitch Pokémon and using them throughout the land, and the arbitrary code execution possible with the right sequence of actions, there are many great times to be had.
Those are a few from me!