TITLE: Is the Euler–Mascheroni constant an EL-number?
QUESTION [7 upvotes]: This question is based on Chow - What is a closed-form number?.
The author of the linked paper had proposed a plausible definition of "elementary numbers" (which he calls "EL-numbers") concept, building it analogously to the concept of elementary function (author admits, "elementary number" would be a better name for his proposal, but the term is already occupied).
So, the author defines a set of $\mathbb{E}$ of "EL numbers" which stands for "elementary" and well as "exponentially-logarithmic".
He defines the set as any numbers that can be produced by applying finite number of field operations, exponential and logarithmic functions to the number $0$.
For instance, in his system
\begin{gather*}
1=\exp(0) \\
e=\exp(\exp(0)) \\
i=\exp\left(\frac{\log(-1)}2\right)=\exp\left(\frac{\log(0-\exp(0))}{\exp(0)+\exp(0)}\right) \\
\pi=-i\log(-1)=-\exp\left(\frac{\log(0-\exp(0))}{\exp(0)+\exp(0)}\right)\log(0-\exp(0)).
\end{gather*}
It turns out that any root of a polynomial with rational coefficients, expressible in the radicals, is also in $\mathbb{E}$.
So, my question is, does the Euler–Mascheroni constant $\gamma$ belong to the EL-numbers? I think no, but that it is "nearly-elementary" in the same way as the digamma function is a nearly-elementary function.
My thoughts on this revolve around these points:

There is symmetry between $\pi/4$ and $e^{-\gamma}$

$\gamma=\psi(1)$, but $\psi(x)$ is antidifference of $1/x$ while logarithm is antiderivative. $\psi(x)$ relates to $\log x$ the same way as Bernoulli polynomials relate to monomials (both Bernoulli polynomials and $\psi(x)$ are slices of Hurwitz Zeta function).

Many divergent integrals and their logarithms regularize to $\gamma$, particularly, $\operatorname{reg} \int_0^1 \frac1x dx=\gamma$ (which makes it in some sense the regularized value of logarithm at zero).

REPLY [10 votes]: Despite the simplicity and elegance of the definition of EL numbers, it is very hard to prove that an explicit number is not EL, as Chow points out throughout the paper.
In particular, I am certain that as of right now, nobody knows if the Euler-Mascheroni constant is or isn’t EL. It most likely isn’t, but we can’t even prove that it is irrational.