TITLE: An amazing approximation of $e$
QUESTION [10 upvotes]: As we can read in Wolfram Mathworld's article on approximations of $e$, the base of natural logarithm,

An amazing pandigital approximation to e that is correct to $18457734525360901453873570$ decimal digits is given by $$\LARGE \left(1+9^{-4^{6 \cdot 7}}\right)^{3^{2^{85}}}$$ found by R. Sabey in 2004 (Friedman 2004).

The cited paragraph raises two natural questions.

How was it found? I guess that Sabey hasn't used the trial and error method.
Using which calculator can I verify its correctness "to $184\ldots570$ decimal digits"?

REPLY [15 votes]: $$\begin{aligned}
(1+9^{-4^{42}})^{3^{2^{85}}}
&=(1+9^{-4^{42}})^{3^{2*2^{84}}}\\
&=(1+9^{-4^{42}})^{9^{2^{84}}} \\
&=(1+9^{-4^{42}})^{9^{4^{42}}}\\
&=\Bigl(1+\frac1{9^{4^{42}}}\Bigr)^{9^{4^{42}}}\qquad\text{where }=\left(1+\frac1n\right)^n.
\end{aligned}$$
This is just the limit definition of $e$ with a large number as an approximation for $\infty$.
Edit:
Numberphile just did a video on this, which also gives a pandigital approximation for $\pi$, but it's only accurate up to ten digits.