Circle and square go hand in hand!

Ratio of the areas of the inscribed circles is the same as the ratio of the areas of the squares. So only squares need to be considered.

Dealing with only ratios of sides, one of the sides can be arbitrarily set. Let the first, largest, square have a side 1.

The square inscribed in it by connecting midpoints will have sides $\sqrt 2$ in length, so its area will be $\dfrac12$ of the original area.

Each inscribed square will have $\dfrac12$ of the area of the one it is inscribed into.

There are total of $21$ squares, which means the area is shrunk altogether $20$ times down to $\dfrac{1}{2^{20}}$ .

The ratio of the largest to the smallest is then

$\dfrac{1}{\frac{1}{2^{20}}}=2^{20}=\boxed{1 048 576}$