(Too long for a comment)

I don't know if there's a simpler form, but the sum of factorials has certainly been well-studied. In the literature, it is referred to as either the *left factorial* (though this term is also used for the more common subfactorial) or the *Kurepa function* (after the Balkan mathematician Đuro Kurepa).

In particular, for $K(n)=\sum\limits_{j=0}^{n-1}j!$ (using the notation $K(n)$ after Kurepa), we have as an analytic continuation the integral representation

$$K(z)=\int_0^\infty \exp(-t)\frac{t^z-1}{t-1}\mathrm dt,\quad \Re z>0$$

and a further continuation to the left half-plane is possible from the functional equation $K(z)-K(z-1)=\Gamma(z)$

An expression in terms of "more usual" special functions, equivalent to the one in Shaktal's comment, is

$$K(z)=\frac1{e}\left(\Gamma(z+1) E_{z+1}(-1)+\mathrm{Ei}(1)+\pi i\right)$$

where $E_p(z)$ and $\mathrm{Ei}(z)$ are the exponential integrals.

The sum of squares of factorials does not seem to have a simple closed form, but the sequence is listed in the OEIS. One can, however, derive an integral representation that could probably be used as a starting point for analytic continuation. In particular, we have

$$\sum_{j=0}^{n-1}(j!)^2=2\int_0^\infty \frac{t^n-1}{t-1} K_0(2\sqrt t)\mathrm dt$$

where $K_0(z)$ is the modified Bessel function of the second kind.