I didn't have all too many Game Boy games growing up, and I only began playing with the release of the Game Boy Advance. In any case, the first games that come to mind are the Pokémon games, as many have already said. My favorite of the series has to be Fire Red, but I was a huge fan of Emerald as well. I actually played all of the variations but those were my favorite.
Besides the Pokémon series I recall enjoying the Mega Man Battle Network series, as well as obsessively playing Super Mario Advance 4, which was the GBA remake of Super Mario Bros. 3.