In front of me, I have an arbitrary number of four-sided, six-sided, eight-sided, ten-sided, twelve-sided, and twenty-sided dice. Using any number and combination of these, is it possible to exactly duplicate the results of a seven-sided die?
Is it possible to duplicate a seven sided die without rerolling any dice? If not, what is the proof? If it is possible, what combination allows this?
Asymptotically, yes:
Form seven sets of extremely large numbers of equivalent sets of dice, for instance seven sets each having a quadrillion 20-sided dice. Call the sets $1, 2, \ldots, 7$. Roll your seven quadrillion dice and compute the seven totals. Whichever set has the highest total tells you which of the seven sides of the desired die appeared.
You can make the chance that two such "winning" sets have the exact same value arbitrarily small by increasing the total number of dice in each set.
But there will always be a (vanishingly small) chance the two "winners" have the same value.