jmc

To express the number $0.\overline{7}$ as a fraction, we call it $x$ and subtract it from $10x$: \begin{array}{r r c r@{}l} &10x &=& 7&.77777\ldots \\ - &x &=& 0&.77777\ldots \\ \hline &9x &=& 7 & \end{array} This shows that $0.\overline{7}=\frac{7}{9}$.

To find the reciprocal, we just switch the numerator and denominator: $1/0.\overline{7}=\boxed{\frac{9}{7}}$.