According to an article from RPGamer, there was a fourth Final Fantasy game in production for the Famicom. Scroll down a bit and you can see a paragraph about it and a scan from an old issue of Famitsu (note that only the large picture of the game in the scan is a picture of "Final Fantasy IV" (as this game was originally going to be called). The other, smaller picture is of Final Fantasy III). BugCatcher translated the scan, and this is what it says:
Every time a new game in the Final Fantasy series is released, it uses a new system. In the latest, IV, as well, a new system is introduced that is guaranteed to surprise players.
For better or worse, the dungeons are long, as expected. However, save points have been placed inside. Save junkies will no doubt wander everywhere.
The popular job class is back, with even more classes. Magician, priest, cook, and carpenter, among other jobs, make an appearance; expect the unexpected. The battle system is definitely Final Fantasy as well; you can release doves that blind enemies, use prayer to make them die, and more.
Vehicles are back as well, and it looks like there is finally an Airship job class. All previous airships can be bought, plus a new, gigantic battleship which fills up 1/3 of the screen. Townspeople will also buy and use airships, so you may get into a mid-air collision. Sadly, you are not the sole ruler of the skies anymore.
FF's style--a unique battle system, cute characters, and a wealth of items--are back in 'IV'" .
"Predictions From Competitors
Airships will be high-speed machines with onboard nuclear weapons that transform into giant robots in midair. They will be known as Chocobo and Super Chocobo. Summon magic will better as well; for example, Mega Flare will become Giga Flare. Dungeons will be tougher too! - Enix
The Final Fantasy series uses one original system after the next, so it's a game that won't lose the player's interest, unlike Dragon Quest. Even so, I want them to put a lot of effort into it. For example, some special weapons, variations in the vehicles and so forth. - Namco"
"There is an Airship Shop in this town. An array of surviving Airships from the past are there."
BOTTOM LEFT CAPTION
"Fight scene from III. There are plenty of things in the game that will bring back memories."
"These airships are part of the many vehicles in Final Fantasy. In IV, the Airship Shop finally makes its appearance. Famous airships straight out of II and III are sold here."
Some of those ideas sound pretty darn interesting... too bad they never actually made it into any actual FF game. Then again, who knows? 8-)