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Best answer as chosen by Asker / Ratings
There are a number of places you can get strenght spheres around Spira while going through the game.
You can get one for winning the Blitzball game against the Luca Goers during the Story.
If you go back to the temples where you get each of the Aeons you will get a number of special spheres to raise stats by 4. To get a strenght sphere go back to the chamber of the fayth where you get Yojimbo. You can also get some strenght spheres by dodging lightning bolts in the Thunder Plains.

There is a way to get an ulimited amount of each of the special stat spheres. When you catch fiends for the Calm Lands arena the gut there creates new monsters there is one list of monsters he creates called species conquest and all the monster he creates for that list drop the special stat spheres when you defeat them. It takes a while to get each of these fiends in the species conquest list because you have to catch the max amount of a fiend of the same species from each area an example would be the small bird fiends when you catch most of them from the areas they are in then the guy in the Calm Lands arena will create the special species conquest fiend of that type. Its best if you just catch the max amount of each fiend from every area.