J'ai acquis bon nombre de pierres toutes plus belles les unes que les autres.
Does this phrase basically mean that the multiple stones the speaker has obtained each have different levels of beauty?
No, it just means the stones are very pretty.
The expression tou(te)s plus X les un(e)s que les autres is just another way to say très X. The literal meaning is kind of a contradiction — in your example, it would mean that when you looked at any one stone closely, you'd always find it prettier than all the other stones; but the stones can't all be the prettiest stone. This contradiction makes the expression a little tongue-in-cheek; the literal meaning is not intended when using this expression.