It seems apparent that all contractible simplicial complexes have Euler characteristic of $\chi=1$. Are there any non-contractible (path-connected) simplicial complexes with Euler characteristic $\chi=1$?
Yes, many. For a concrete example, how about an octahedron with an edge running through the middle?
It's worth noting that such a complex must necessarily have dimension at least 2, since a path connected 0-complex is just a point, and a noncontractible path-connected 1-complex is homotopy equivalent to a wedge of circles, which has nonpositive Euler characteristic.