jmc

We can write $\frac{123}{999}$ as $0.\overline{123}$. To see why, let $x=0.\overline{123}$, and subtract $x$ from $1000x$: \begin{array}{r r c r@{}l} &1000x &=& 123&.123123123\ldots \\ - &x &=& 0&.123123123\ldots \\ \hline &999x &=& 123 & \end{array} This shows that $0.\overline{123}=\frac{123}{999}$.

This decimal repeats every 3 digits. Since $123,999$ is divisible by $3$ (as the sum of the digits of $123,999$ is equal to $33$), it follows that $123,999^{\mathrm{th}}$ digit after the decimal point is the same as the third digit after the decimal point, namely $\boxed{3}$.