TITLE: How do you calculate $ 2^{2^{2^{2^{2}}}} $?
QUESTION [21 upvotes]: From information I have gathered online, this should be equivalent to $2^{16}$ but when I punch the numbers into this large number calculator, the number comes out to be over a thousand digits. Is the calculator wrong or is my method wrong?

REPLY [4 votes]: Your equation can be simplified using Knuth's up arrow notation.
\begin{equation*}
2^{2^{2^{2^2}}} = 2 \uparrow\uparrow 5
\end{equation*}
(because we can calculate tetration with Knuth's up arrow notation)
By definition of Knuth's up arrow notation, You can get this result.
\begin{equation*}
2\uparrow\uparrow5 = 2^{(2^{(2^{(2^2)})})}
\end{equation*}
And according to web2.0calc,
\begin{equation*}
2^{(2^{(2^{2})})} =  65536
\end{equation*}
Finally, the answer would be:
\begin{equation*}
2^{65536}
\end{equation*}
(correct me if I'm wrong, this was my first answer on Mathematics SE)