The problem is not well defined, because it is not clear what exact relation between
the series and "an expression" are you looking for. The series is divergent for
all $x\neq 0$. However:

It is easy to see that $y(x)=\sum_{n=0}^\infty n!x^n$ satisfies the differential
equation
$$x^2y'+(x-1)y+1=0,$$
and the initial condition $y(0)=1$. This is a linear first order differential equation, which can be solved explicitly:
$$y(x)=x^{-1}e^{-1/x}\left(\int_x^\infty t^{-1}e^{1/t}dt+c\right).$$
Unfortunately, $c$ is arbitrary here, but your series is an asymptotic series at $0$
for all these solutions. Or "formal Taylor series", if you wish. So in some sense your series "represents" all these solutions.