Mathematical constants $\pi$, $e$, $i$ have a lot of algebraic roles. They appear as identity elements, idempotents, invariant elements etc against various operations and sets.
This is illustrated by the following identities:
$$(e^{i\pi})^2 = 1$$
$$i^4=1$$
$$(e^x)'=e^x$$
I wonder, what is the algebraic role of Euler-Mascheroni constant, $\gamma$?
The constant manages to find its way to a number of important places like: infinite product, improper integral, definite integral of transcendental functions, power series, continued fractions, etc. To quote a few "identities" involved $\gamma$ we have:
$\displaystyle \int_{0}^1 \left(\dfrac{1}{\ln x} + \dfrac{1}{1-x}\right)dx = \gamma$
$\displaystyle \int_{0}^\infty e^{-x}\ln^2 x dx = \gamma^2 + \dfrac{\pi^2}{6}$
$\gamma = \displaystyle \sum_{n=2}^\infty (-1)^n\dfrac{\zeta(n)}{n}$