smc

Taking successive exponentials ${\log }_2({\log }_2(x))=2^2=4$ and ${\log }_2(x)=2^4=16$ and $x=2^{16}$. Now $2^{10}=1024\approx 10^3$ and $2^6=64$ so we can approximate $2^{16}\approx 64000$ which has 5 digits. In general, $2^n$ has approximately $n/3$ digits. $\boxed{5}$