**A**
181 = sqrt( sqrt(sqrt(sqrt(2^{5!}))) - 7 ).

The main point of this puzzle is the fact that 2^{15}-7
is a perfect square. This fact was noted at least as early as 1913,
when Ramanujan asked whether 2^{n}-7 can be a square
for any integer n>15. That this never happens again
was finally proved two generations later by Nagell
(*Arkiv för Matematik* **4** (1960), 185-187).

This curiosity shows up at least twice in research mathematics:
in arithmetic algebraic geometry,
it lets one show that a certain model of the Klein quartic curve mod 2
is the unique curve of genus 3 with the maximal number of points
over the field of 2^{13} elements
(see my article on the Klein quartic in
“The Eightfold Way”); and in coding theory,
it seems to promise a perfect 2-error-correcting binary code
of length 91, which unfortunately cannot exist
due to a further integrality criterion (see for instance
MacWilliams and Sloane's book on error-correcting codes).