Question: Fifty years ago, we were taught to calculate square roots one digit at a time with pencil and paper (and in my case, an eraser was also required). That was before calculators with a square root button were available. (And a square root button is an incredibly powerful doorway to the calculation of other functions.)

While trying to refresh that old skill, I found that the square root of 0.1111111.... was also a repeating decimal, 0.3333333.... Also, the square root of 0.4444444..... is 0.66666666..... 

My question: are there any other rational numbers whose square roots have repeating digit patterns? (with the repeating digit other than 9 or 0, of course)? The pattern might be longer than the single digit in the above examples.

Answer: 
Just take your favorite rational number with a repeating decimal:

$$\frac{1}{11} = .09090909\ldots.$$

Square it:

$$\frac{1}{121} = .00826446280991735537190082644628099173553719\ldots,$$

and you've created an example. This one has period 21 and its square root has period 2.
Any repeating digit pattern *must* be rational, so take a rational number $\frac{p}{q}$, and square it to get $\frac{p^2}{q^2}$, and there you have it!

It's worth noting that, in the examples you gave, $\sqrt{0.1111...} = \sqrt{\frac{1}{9}} = \sqrt{\frac{1}{3^2}} = \frac{1}{3}$, and $\sqrt{0.44444...} = \sqrt{\frac{4}{9}} = \sqrt{\frac{2^2}{3^2}} = \frac{2}{3}$.