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Is Banach-Tarski* a valid answer? (Kleene star.)

No, but (Banach-Tarski)(Banach-Tarski)* would be, or possibly (Banach-Tarski)+ depending on your system. You can't get to zero copies.

Wikipedia says you can collapse two balls into one, so doesn't that mean you can cut your ball in half, mold the halves into balls, and then merge those two half-volume balls into one half-volume ball? And if you can do that, can't you get to an infinitely small ball?

Arbitrarily small, yes, but never zero. You will always have c points, where c is the cardinality of the continuum.

Xeno would agree!

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