Four fat kids fall down a hole, somehow become the heroes of the world. That's... basically the entire story of Final Fantasy III. But that sells it all so short.
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I recently played through the Pixel Remaster versions of the first three Final Fantasy games, which includes the first 2D version of Final Fantasy III released outside of Japan.
The first two are classics in their own right, but they're both clearly limited by their engine and scope. The first in particular feels really almost comically small in comparison to the games that come right after it and is as much a proof of concept for a series as it is a standalone game. Final Fantasy II plays with heavy alterations to stat progression and magic, as well as an increased focus on story, but the actual scope and ambition of the game still feels constrained by what the first did. There is a relatively clear delineation between towns, dungeons, and overworld, and despite a focus on story the actual storytelling is pretty light, relying mostly on environmental storytelling and the occasional exposition dump--real scenes are very very minor affairs. Interactivity in towns and dungeons is basically unheard of.
FFIII looks and plays a lot like its two predecessors but there is so much more going on from the word go. The environments are interactive, with switches for the characters to hit and healing fountains to drink out of. Hidden items to be found around towns. Pianos to play. NPCs that do more than wander around aimlessly. And despite the pull back on story focus from II, this actually doesn't really mean much in practice, as the actual amount of scenes is greater and they are generally much more animated and interesting to watch. Scenario design is also much more varied, a couple times you are required to shrink yourselves to enter a dungeons, at one point your airship is shot down and you're trapped in a kingdom fighting amongst itself, another time you get captured. FFII played with this more than the first, but FFIII turns this up a considerable notch. There is just so much more to physically do in this game.
And so many Final Fantasy staples got their start here. The true job swapping system originates here. Unique battle commands outside of magic originate here (like Thieves being able to steal, and Dragoons' ability to jump). Moogles first appear here, as does the Fat Chocobo. Summon magic proper appears for the first time with most of the stock standard summons--Ifrit, Shiva, Ramuh, etc. It's hard to overstate how influential it is on the future of the series.
The big word that comes to mind when discussing Final Fantasy III in comparison to its predecessors is ambition. Not content with just a canoe, ship, and airship, Final Fantasy III has the party getting three different airships throughout the game, each with their own abilities, two of which the party has access to at the same time at the end of the game. And rather than just one overworld, FFIII reveals that the initial overworld you begin the game on is actually a small floating island hovering above yet another fully featured overworld. And that's not even getting into the ability to have one of your airships go underwater revealing another area to explore with its own (albeit limited) secrets. On top of all that there are actual optional hidden dungeons scattered around the overworlds with their own tricks and challenges to overcome--every dungeon in the first two is required to beat their respective game, but FFIII has not just a couple but several optional areas you never have to explore. The sheer scale of the world is far larger and grander than the first two titles dreamed of.
I think it was actually a great shame the game was not released outside of Japan in its original time frame, as it is little wonder it holds such an important place in the fandom's hearts in its home country. I kind of wonder what its reception in the West would be had it released on the NES like originally intended.
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As an aside for those of you who have only played the 3D remake that first released on the Nintendo DS (which up until the Pixel Remaster was the only version ever released outside of Japan), I highly recommend playing the Pixel Remaster version. There were a lot of game design concessions that the 3D remake had to make to fit it in the DS hardware that make for a lesser experience, turning what was originally a quick and snappy RPG into a slow slog. The Pixel Remaster also drops the CP requirements for job switching from the NES version as well as the Job adjustment period from the 3D Remake making job swapping clean and simple.