The bound given by F. Luca and cited in Gjergji Zaimi's answer has been recently improved, although only by a factor $\log \log \log n$. Precisely, in *C. Sanna, On the sum of digits of the factorial, J. Number Theory 147 (2015), 836--841*, the author proved that
$$\min\{s_b(n!),s_b(\Lambda_n)\} > c_b \log n \log \log \log n , $$
for all integers $n > e^e$, where $s_b(\cdot)$ is the sum of base-$b$ digits function, $c_b > 0$ is a constant depending only on $b$, and $\Lambda_n$ is the least common multiple of $1, \ldots, n$. Clearly, this inequality also holds with $s_b$ replaced by $t_b$, i.e., the function counting the number of non-zero base-$b$ digits of its argument, since $t_b(m) \geq s_b(m)/b$ for each positive integer $m$.