Question: The exponential integral function $x \mapsto E\_1(x)$ is strictly decreasing on the positive real
axis and, so, is globally real analytically invertible there. Where can I find information concerning
a maximal domain in the complex plane over which $E\_1$ possesses a holomorphic global inverse that continues the real one?

Answer: 
Just two remarks: a) it is simply related with the integral logarithm,
$$\int\_{-\infty}^x\frac{e^t}{t}dt=\int\_0^{e^x}\frac{dt}{\log t},$$
which is a better known function. b) There is a whole book on it:

Niels Nielsen, Theorie des Integrallogarithmus und verwandter Transzendenten. 37.0454.01
Leipzig: B. G. Teubner. VI u. 104 S. (1906).
Just a remark: for all your elementary needs regarding special functions, the one-stop shop is the [Digital Library of Mathematical Functions](https://dlmf.nist.gov/).

There is an [index](https://dlmf.nist.gov/idx/), and you find what you are looking for [here](https://dlmf.nist.gov/6.2#i).