Is there any finite 8-dimensional point group which contains the $E_8$ Coxeter group as a subgroup other than $E_8$ itself?
Let $G$ be a finite subgroup of $GL_n(\mathbb R)$ containing $W(E_8)$ as a subgroup. Because $G$ is compact, $G$ must preserve a symmetric positive definite form on $\mathbb R^8$. Since $W(E_8)$ preserves a unique such form, it must be that one.
Let $H$ be the largest subgroup of $G$ generated by reflections. Then $H$ contains $W(E_8)$ and thus is an irreducible reflection group, hence a Coxeter group. Examining the table of Coxeter groups and looking for entries in dimension $8$, there are four possibilities: $W(A_8), W(B_8), W(D_8), W(E_8)$. Because $W(E_8)$ has the highest order of these, we must have $H = W(E_8)$.
Now, by construction, $H$ is a normal subgroup of $G$, so $G$ normalizes $W(E_8)$, and hence $G$ is contained in the automorphism group of the $E_8$ root system. Because the Coxeter-Dynkin diagram $E_8$ has no nontrivial automorphisms, this is $W(E_8)$ and so $G= W(E_8)$, as desired.