Must defeat the Ultimate Weapon at Cosmo Canyon. (disc 3)
The Ultima Weapon will increase in power according to how many Hit Points Cloud has. If he has a low amount of HP, the outer edges of the sword will turn from white, to blue.
Barret's Missing Score:
If you have Barret in your party, you'll come across it while climbing the stairs to reach Professor Hojo during the Midgar Invasion. (disc 2) You must get it during the Midgar Invasion because if you don't, there is not another chance to get it.
The Missing Score will increase in power according to the amount of AP that's equipped with the weapon. So, for instance, you would want to put a mastered Knights of the Round materia in one of the slots because it took alot of AP to master it.
Cid's Venus Gospel:
After the Rocket is launched, go to Rocket Town and talk to the old man sitting in a chair outside of one of the shops. Keep talking to him and he'll give you Cid's final weapon.
The Venus Gospel will increase in power according to how much MP Cid has. (Hint: This is one of the only times in the game where the HP--MP materia will come in handy!)
Tifa's Premiem Heart:
Go to the excavation site on the northern continent. Excavate for normal treasure and have the workers dig right at the tip of the long metal piece near the entrance. You'll get the Key to Sector #5. Go to Midgar, enter the gate, and go to the Wall Market. If you enter the item shop, the computer will work and you'll recieve the Premiem Heart.
The Premiem Heart will increase in power when one of her limit break slots gets a "miss". (However, I found that whenever you win a battle, the Premiem Heart doubles its damage power... but think what you will.)
Aeris's Princess Guard:
In the Temple of the Ancients, if you turn the time clock to the IV position, a path will lead you to a room where the Princess Guard lies.
The Princess Guard will increase in power according to the weakness of Aeris's allies. If their points are lower, she will hit harder.
Cait-Sith's HP Shout:
During the Midgar Invasion (disc 2), go up to Shinra HQ. Enter, and take the elevator to the 64th Floor. Go to the locker room and open a locker in the middle of the top row of lockers. HP Shout will be inside.
(Unknown)
Nanaki's Limited Moon:
During disc 3, revisit Cosmo Canyon with Nanaki in your party. A man at the gate will inform you that Bugenhagen isn't feeling well. Go up to Bugenhagen and watch the dialog unfold. Bugenhagen will die, but he will give Nanaki his Limited Moon.
(Unknown)
Vincent's Death Penalty:
In disc 2, take the submarine below the ocean surface and travel southward along the wall. You will eventually come to a small spot with a cave. Enter the cave and when you reach the end, ascend to the surface. Get off the submarine, put Vincent in your party, and go behind the waterfall. Watch the story. You will leave with nothing, but you must do this first. Now, in disc 3, go back to that same place behind the waterfall and search the bright light. Not only will you recieve Death Penalty, but you will also recieve Vincent's Final Limit Break, Chaos.
The Death Penalty will increase in power according to how many enemies Vincent kills with the weapon. In other words, the more enemies he kills, the more power he'll have.
Yuffie's Conformer:
In the Sunken Shinra Plane, the Research Room (first door by the save point) has a treasure hidden from view behind a pipe. Inside rests the Conformer.
The Conformer will increase in power according to the level of Yuffie's enemy. The higher level an enemy has, the more powerful attack the Conformer will perform. (Hint: If you equip Yuffie with the Conformer and the Morph materia, she will hit extremely hard, making it easy to Morph enemies into items.)