Recently I've found a polyhedron with Euler characteristic $\chi=9$. This is made from the Octahemioctahedron with adding the intersections of the hexagon faces as vertices. It has $V=13, E=36, F=32$.
I know, that Euler characteristic is in close connection with simply connectedness: $\chi<2$ means not simply connectedness. But what does $\chi>2$ mean? Is my polyhedron simply connected or not?
Your polyhedron is not a topological surface, by the classification of closed surfaces: An orientable surface of genus $g$ has Euler characteristic $2 - 2g$; a non-orientable surface has genus $1 - g$ for some $g \geq 0$. In all cases, $\chi \leq 2$.
Based on the diagrams in your other question, your polyhedron is:
Obtained from eight regular tetrahedra by identifying one vertex from each, then gluing edges in pairs.
Obtained from a torus by dividing each hexagonal face into six equilateral triangles, then identifying the four central vertices of the hexagons and gluing the radial edges in pairs.
In either picture, the fact that some edges meet four faces shows the polyhedron is not a surface. The first description makes it clear you've got (a polyhedron homotopy equivalent to) a bouquet of eight $2$-spheres, which is simply-connected.