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The largest truely round base plates in the LEGO system fit in a 8x8 square. You can build them from two LegoPlate, Round Half 4 x 8 or from four quarters.
For a bigger (and thicker!) base plate there is a Duplo version of the above mentioned base plate, fitting into a 16x16 studs LEGO system square: LegoDuplo, Plate Round Corner 4 x 8 Double
EDIT: For a ...
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