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).

exponential motivesFresan and Jossen argue that an analogue of Grothendieck period conjecture for exponential periods implies that $\gamma$ is algebraically independent of $2\pi i$. See Corollary 12.8.8 here. $\endgroup$