What is the $111$th digit after the decimal point when $\frac{33}{555}$ is expressed as a decimal?
Using long division, we find that $\frac{33}{555}$ can be expressed as a repeating decimal $0.0\overline{594}$.

After the first digit, there is a three-digit repeating block. We want to find the $110$th digit after the first digit. The remainder when $110$ is divided by $3$ is $2$. Therefore, the $111$th digit is the second digit in the repeating block, which is $\boxed{9}$.