For $x \in \mathbb{R}_{>0}$, the exponential integral is defined as follows: $$\operatorname{Ei}(x) := PV \int_{-\infty}^{x} \frac{e^{t}}{t}dt. $$ Here, $PV(\cdot)$ is the Cauchy principal value.
Several generalizations of this special functions have been considered, including the ones associated with the $E_{n}(\cdot)$ function and this one by Nantomah et al.
I wonder whether the following, nested generlization of the exponential integral has been considered already in the literature. Let $\operatorname{Ei}(1,x) := \operatorname{Ei}(x), $ and define $$\operatorname{Ei}(2,y) := PV \int_{-\infty}^{y} \frac{\operatorname{Ei}(1,x)}{x} dx. $$
We further define every nested exponential integral of subsequent order by the recursive relationship $$\operatorname{Ei}(m,z) := PV \int_{-\infty}^{z} \frac{\operatorname{Ei}(m-1,x)}{x} dx , \tag{*}$$ somewhat like the polylogarithms.
Altough I have found a few double integrals of the exponential integral function before (see for instance p. 618 of volume 1 of Integrals and Series by Prudnikov et al.), I haven't encountered this generalization of the exponential integral before. I wonder what its properties are, and if definite integrals over it can could be obtained. One could do something similar for the logarithmic integral function.
Question: have the nested exponential integral functions as defined in $(*)$ been described in the literature before?