Fun fact: Final Fantasy III was the first game in the series I played to completion (on the Nintendo DS, thankfully: I'm not planning on going anywhere near the Famicom original). That's not to say it's one of my favorites, per se, but I do have some fond memories of this title even if does lack the same level of spectacle and character as later entries.
For starters, this is where the job system got its start! Granted, Final Fantasy V and Tactics both elevated that system to new heights, but III is where it all began, and the simple pleasure of customizing your party of tiny Warriors of Light is a delight. The game's overworld themes are also quite beautiful: "Eternal Wind" is a great piece of music, as is the melancholy "Boundless Sea." And, lest we forget, that grueling final dungeon gauntlet is immensely satisfying to clear... once you get past the fact that you need to complete two entire dungeons and defeat six bosses without saving in order to see the credits roll, and the Cloud of Darkness is no slouch.
...Okay, on second thought, maybe this isn't strictly speaking a "positive" memory. But if you're curious and willing to take on a challenge, Final Fantasy III's DS remake (also available on mobile platforms and Steam) is worth a shot.