Final Fantasy III (not Final Fantasy VI which was originally released as Final Fantasy III in the United States for SNES) was never released on any of Sony's consoles.
The original game was released for Famicom (known as Nintendo Entertainment System in the United States) in Japan only. But there's a fan-translation for it, you can play it on emulators.
The game was remade in 3D graphics for Nintendo DS and released globally for the first time. The same version was later ported with enhanced graphics to Apple's iOS devices such as iPhone and iPod touch. An HD port of it was released for iPad as well. The same enhanced port was recently released on Google Play store for the Android-based smartphones and tablets. In addition to that, this port will also be released for PSP in Japan only soon.
You have all the main Final Fantasy games for PS1, PS2, and PS3. However, you are missing some of the spin-offs and imports. For PS1:
Final Fantasy Tactics
Final Fantasy Anthology European version, which has FF4 & 5 instead of FF 6 & 5
Final Fantasy Collection Japanese version, which has FF4, 5, and 6
Final Fantasy Japanese version, standalone release of FF1
Final Fantasy II Japanese version, standalone release of FF2
Final Fantasy V Japanese version, standalone release of FF5
Final Fantasy VI, standalone version released in Japan and Europe, also includes FF10 demo disc for the PS2
Final Fantasy VII International, this is a Japanese version which has a bonus information disc
These games are related to Final Fantasy:
Chocobo no Fushigi na Dungeon, Japan only
Chocobo's Dungeon 2
Chocobo Racing
Ehrgeiz: God Bless the Ring
Vagrant Story
For PS2, you are missing:
Dirge of Cerberus -Final Fantasy VII-
Dirge of Cerberus -Final Fantasy VII- International, this is a Japanese version with some changes
Final Fantasy X European version, it has a lot of extra stuff not in the American release
Final Fantasy X: International, this is a Japanese version, exactly the same as the European version and has English voice and text as well
Final Fantasy X-2: International + Last Mission, Japanese only, has some extra stuff
Final Fantasy XII International Zodiac Job System, Japanese only, an enhanced version of FF12 with new job system
Related games:
Dragon Quest and Final Fantasy in Itadaki Street Special, this is Japan only
Kingdom Hearts
Kingdom Hearts II
Kingdom Hearts Re:Chain of Memories
Now for PS3:
Final Fantasy XIII Ultimate Hits International, Japan only, has some changes, a booklet and other some stuff
Related games:
Crystal Defenders, this is download only
Not sure if you are also collection Sony PSP games but in case you do, here is what you are the games:
Final Fantasy I remake
Final Fantasy II remake
Final Fantasy IV Complete Collection, it's a 2D remake of FF4 and its sequel "The After Years"
Final Fantasy Tactics: The War of the Lions, enhanced version of FF Tactics
Crisis Core -Final Fantasy VII-, prequel of FF7
Dissidia Final Fantasy, fighting game
Dissidia Final Fantasy: Universal Tuning, enhanced port of Dissidia Final Fantasy, Japan only
Dissidia 012 Final Fantasy, sequel of the other game
Final Fantasy Type-0, Japan only, same world of FF XIII
Dragon Quest & Final Fantasy in Itadaki Street Portable, Japan only
Final Fantasy: The Spirits Within, UMD video
Final Fantasy VII: Advent Children, UMD video