Let
$$
f(a,x) = G_{2,3}^{3,0}\left(x\left|
\begin{array}{c}
1,1 \\
0,0,a \\
\end{array}
\right.\right)
$$
Then it is not hard to determine that, for positive $x$,
$$
f(1,x) = - \mathrm{Ei}(-x)
$$
Additionally, it follows from Mellin-Barnes representation of the Meijer G-function that
$$
f(a+1, x) = a \cdot f(a,x) - x \partial_x f(a,x) = -x^{a+1} \partial_x \left( x^{-a} f(a, x) \right)
$$
This means that
$$
x^{-n} f(n, x) = (-1)^n \frac{\mathrm{d}^{n-1}}{\mathrm{d} x^{n-1}} \left( x^{-1} \mathrm{Ei}(-x) \right)
$$
or
$$
f(n, x) = (-x)^n \frac{\mathrm{d}^{n-1}}{\mathrm{d} x^{n-1}} \left( x^{-1} \mathrm{Ei}(-x) \right)
$$
Verification in Mathematica:
In[38]:= {MeijerG[{{}, {1, 1}}, {{0, 0, 7}, {}},
x], (-x)^7 D[ExpIntegralEi[-x]/x, {x, 6}]} /. x -> 1`16
Out[38]= {1348.4143043955619, 1348.41430439556}