I often encounter this constant in my research, but I wonder what typical roles does it play in other areas of mathematics? Wikipedia mentions probability theory but nothing exact.
Also, I am interested to know best keywords to search about this constant.
This constant appears in Weierstrass factorization of $\Gamma(s)$:
$$ {1\over\Gamma(s)}=se^{\gamma s}\prod_{k\ge1}\left(1+\frac sk\right)e^{-s/k} $$
and also in Mertens' formula:
$$ \prod_{p\le x}\left(1-\frac1p\right)={e^{-\gamma}\over\log x}\left[1+\mathcal O\left(1\over\log x\right)\right] $$
Mertens' formula accounts for most appearance of $e^\gamma$ in number theory. For instance, it can be used to deduce the minimal order of Euler's totient function:
$$ \liminf_{n\to\infty}{\varphi(n)\log\log n\over n}=e^{-\gamma} $$
and the maximal order for divisor sum function (aka Gronwall's theorem):
$$ \limsup_{n\to\infty}{\sigma(n)\over n\log\log n}=e^\gamma $$
Number theory: The geometric mean of prime numbers around $n$, is approximately $\color{red}{e^{-\gamma}}\log n$.
Probability: If $u_1=$ random real number in $(0,1)$, and $u_k=$ random real number in $(0,u_{k-1})$, then the probability that $\sum_{k=1}^\infty u_k>1$, is $1-\color{red}{e^{-\gamma}}$.
(To find more, you can try putting $e^{-\gamma}$ in the search bar here on MSE, or at approach0.)
