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The game maps, in particular the first three generations, were very large and diverse for the consoles they were on. It tapped into your childlike lust for adventure and exploration, making it feel bigger than it actually was.
The monster designs are exceptionally good. There is a diverse selection of monsters to choose from, and there is always a type of Pokémon that appeals to someone. The franchise has an art style that is simple, but allows for detail when needed, and therefore, they are not stuck working with a certain level of detail that needs to be achieved for every design. Tyrantrum doesn't feel out of place when stuck next to Snorlax.
Not only that, but the customizability offered to players was unheard of at the time, and is still one of the main appeals of Pokémon to this day. It's not like picking three generic weapons and going about your day, you have hundreds of unique monsters to discover, catch, train, and grow with. Your Pokémon felt like yours. This is something that is well executed in Pokémon. Evolution is something to work for and rewards growth and bond with your Pokémon.
The music is excellent. Simply put, a good soundtrack is memorable and stands the test of time. It's no wonder so many people have strong memories tied to certain Pokémon tracks.