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Alexandre Eremenko
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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.

Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429