1. Vincent's Ghost
This monster supposedly resides in the basement of the Shinra Mansion, Nibelheim. Have Vincent in your party, revisit the basement and there will be a small chance you will fight Vincent's Ghost.
Rune is a demi-god who hides in the background when you fight Safer Sephiroth. If you defeat Sephiroth too easily and quickly, Rune will appear to provide a greater challenge.
A nice rumor, this one. If you manage to resurrect Aeris, Jenova will send one last monster to destroy you - Jenova*REBIRTH. This is the hardest of the Jenova monsters. But only if you resurrect Aeris...
4. Earth Weapon
This Weapon is cunningly disguised as a mountain! Walk around each mountain, and you're supposed to find it eventually.
5. Onyx Weapon
Thanks goes out to Shane for providing the following details.
Onyx Weapon is underneath the Midgar Swamp. You must breed a silver chocobo and race it to S class. It can then go underwater. Swim under Midgar Swamp and you'll encounter Onyx Weapon.
6. Turqoise Weapon
Apparently Robert Devondale alleges that the Planet has unleashed yet another Weapon, the Turqoise Weapon. Travel around the shores near Costa Del Sol, and you might see a tentacle waving about in the shallow water. Make contact with it, and you'll fight Turqoise Weapon.
This monster is supposedly a huge beast. Breed a silver chocobo, travel underwater to the crater, and inside you will fight Alpha-Leviathan. Defeat him, and you'll win the Super Nova materia.
A mysterious monster named Ticondo exists, according to Eecetera@aol.com. To fight Ticondo, you must have 99 tissues in your inventory. Fight at the Battle Square in Gold Saucer, and the counter lady will direct you to fight Ticondo. If you win, you will receive a secret materia called 'Gigo'. But be warned, this monster has 5 million HP.
Oh boy... what's next, fighting Aeris? :) Heh, anyway... Eecetera@aol.com alleges Bugenhagen too can be fought. To confront Bugenhagen, go to Cosmo Canyon and visit the observatory. On your first visit in disk 1, touch the switch leading to the area holding the Huge Materia, and Bugenhagen will say 'Only I can use it'. Continue to use it, and after 20 minutes he will grow angry and fight you.
10. Test 0
In the Japanese version of Final Fantasy VII, this beast is allegedly an optional boss that exists in the Corel Prison, specifically the pit where the empty chest is. It resembles a dog and is supposed to be worth a lot of experience points. However, I haven't been able to find any evidence of this monster's existance either in K. Megura's import FAQ (see Game Information) or in official Japanese game guides. Very suspicious...
UPDATE (July 2006): So it seems that an unofficial guide, The Completely Unauthorized Final Fantasy VII Ultimate Guide (known to hardcore FFVII collectors as the "Versus Guide", after the book's publisher), has some information about this monster. It claims that it did appear in the original Japanese version of FFVII and was removed for the game's English-language release. This book's small section on Test 0, including stats and screenshots, can be found here. Special thanks to Fox Fire for the info and the scan!