Can it be shown that a positive fraction of the base$b$ digits of n! are nonzero (in the limit as $n\to\infty$)?
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I believe the best current lower bound on this is the one given by F. Luca in "The Number of NonZero Digits of n!" Canad. Math. Bull. 45(2002), 115118. It is proven there that the number of nonzero base $b$ digits grows at least as fast as $C_b\log n$. 

