For example, with two squares you can achieve up to 10 regions by choosing the right size and position of your squares.

I agree with Dej Mar and his formula nm(m-1)+2

I see that Charlie just edited his recursive definition anyway.

If we make the n-gons congruent and concentric, we can also arrange the m of them with equal rotations to get a nice symmetrical star shape.

The star has n*m points, one for each corner of each polygon.  The tip of each point is a kite.  Spiraling inward from these kite are more kites (depending of m) with isosceles triangles bordering an (n*m)-gon as the central region.

If you consider each of the n*m spiral arms you can see that the number of kites increases by one if m increases by one and that the number of kites + triangles in an arm is m-1

So the total number of kites and triangles is (n*m)(m-1)
Add the inner polygon and outer region to get the complete formula.