Generally speaking, $\chi$ isn't bounded, and it can be negative or positive. For example, $\chi$ for a double torus is $-2$.
For planar graphs, $\chi$ is always 2. See this page for several proofs of this fact.
$\chi$ for a compact surface is given by the formula:
$$\chi = 2(1 - g)$$
Where $g$ is the genus of the surface, or more intuitively, the number of holes in it. This shows that $\chi$ can be decreased as much as we want by increasing the number of holes in a compact surface.
On the other hand, $\chi$ for $n$ disconnected spheres is $2n$. So similarly, we can increase $\chi$ as much as we want by increasing the number of spheres.
See Wikipedia's article for many more examples.