At the present time, we do not even know how to prove that the Euler-Mascheroni constant $\gamma=\lim_{n\to\infty} \sum_{k=1}^n\frac{1}{k} - \log n$ is irrational, much less transcendental; although it is conjectured to be transcendental. The reason you won't find a lot about this topic online (or in the research literature) is because so little is known about the algebraic properties of $\gamma$.

There is an analgoue of the Euler-Mascheroni constant for Carlitz modules, and I know that there are many transcendence results in the Carlitz (and Drin'feld and T)-module universe that aren't currently provable over number fields. See for example

Tensor Powers of the Carlitz Module and Zeta Values,
Greg W. Anderson; Dinesh S. Thakur,
*The Annals of Mathematics*, 2nd Ser., Vol. **132**, No. 1. (Jul., 1990), pp. 159-191.

But I don't know if irrationality (or transcendence) has even been proven in that setting.