Just to add one more nightmare to marty cohen's list.
For $n=6$
$$\frac{\sqrt{\pi }}{12} \,
_0F_4\left(;\frac{2}{3},\frac{5}{6},\frac{7}{6},\frac{4}{3};\frac{1}{46656}\right)+\Gamma \left(\frac{7}{6}\right) \,
_0F_4\left(;\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{5}{6};\frac{1}{46656}\right)-$$ $$\frac{1}{9} \Gamma \left(-\frac{2}{3}\right) \,
_0F_4\left(;\frac{1}{2},\frac{2}{3},\frac{5}{6},\frac{7}{6};\frac{1}{46656}\right)-\frac{1}{108} \Gamma \left(-\frac{1}{3}\right) \,
_0F_4\left(;\frac{5}{6},\frac{7}{6},\frac{4}{3},\frac{3}{2};\frac{1}{46656}\right)-$$ $$\frac{1}{864} \Gamma \left(-\frac{1}{6}\right) \,
_0F_4\left(;\frac{7}{6},\frac{4}{3},\frac{3}{2},\frac{5}{3};\frac{1}{46656}\right)+\frac{1}{720} \,
_1F_5\left(1;\frac{7}{6},\frac{4}{3},\frac{3}{2},\frac{5}{3},\frac{11}{6};\frac{
1}{46656}\right)$$ which is $\approx 1.56900$
I may have a mistake somewhere since the values I obtained are
$$\left(
\begin{array}{cc}
n & \int_{0}^{\infty}e^{-(x^{n}-x)}\,dx \\
2 & 1.73023 \\
3 & 1.57661 \\
4 & 1.55602 \\
5 & 1.55968 \\
6 & 1.56900 \\
7 & 1.57924 \\
8 & 1.58899 \\
9 & 1.59786 \\
10 & 1.60582 \\
11 & 1.61291 \\
12 & 1.61924
\end{array}
\right)$$ showing a minimum