How many digits will $5^{4^{3^2}}$ produce? According to Wolfram Alpha, $5^{4^{3^2}}$ evaluates to an integer with $183\,231$ digits. How does one find out how many digits such a calculation will produce?

A number $n \in \mathbf N^+$ has $\lfloor\log_{10}n\rfloor+1$ digits. For $n = 5^{4^{3^2}}$, we have 
\begin{align*}
  \log_{10} 5^{4^{3^2}} &= 4^9 \cdot \log_{10} 5\\
           &= 262\,144 \cdot \log_{10} 5\\
           &\approx 183\,230.8
\end{align*}
So $n$ has $183\,231$ digits.
