Digit numbers $\times 2$

Question

Some cute results have every digit doubled.

\begin{align} 99225500774400 = {} & \frac{40!}{31!} \\[8pt] 33554433 = {} & 2^{25} +1 \\[8pt] 222277 = {} & -22^{2^2}+77^3 \\[8pt] 8811551199 = {} & 95^5 + 64^5 \\[8pt] 7755660000 = {} & 95^5 + 65^4 \\[8pt] 334444448888 = {} & 6942^3 - 10^8 \\[8pt] 11881133 = {} & 26^5 - 3^5 \\[8pt] 0.0011223344556677\ldots = {} & \frac 1 {891} \\[8pt] 22116600446655008888446677444 = {} & 148716510336462^2 \\[8pt] \end{align}

The last is from A079036. What are other cute examples?

Answer 1

$$\frac15=0.001100110011\ldots_2\;,$$

$$\frac1{20}=0.001100110011\ldots_3\;,$$

and generally

$$\frac1{101(10-1)}=0.001100110011\ldots$$

in all bases.

Answer 2

$$\frac{2244}{9999}=0.\overline{2244}$$ $$\frac{22446688}{10^8-1}=0.\overline{22446688}$$ $$\frac{112233445566778899}{10^{18}-1}=0.\overline{112233445566778899}$$ $$\frac{2}{909}=0.\overline{0022}$$ $$\frac{1133}{9999}=0.\overline{1133}$$

$$\frac{1}{11}=0.\overline{09}$$