(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.
Best Answer
There is no formula for the nth partial sum of the harmonic series, only approximations. A well known approximation by Euler is that the nth partial sum is approximately $$\ln(n) + \gamma $$ where $\gamma$ is the Euler–Mascheroni constant and is close to $0.5772$. The amount of error in this approximation gets arbitrarily small for sufficiently large values of $n$.
A well known fact in mathematics is that the harmonic series is divergent which means that if you add up enough terms in the series you can make their sum as large as you wish.