jmc

If we multiply this fraction by 10, we do not change any of the digits; we simply shift all the digits one place to the left. Therefore, if we multiply the fraction by a sufficiently large power of 10, then the first digit obtained when dividing by 129 will be the first nonzero digit to the right of the decimal point of $\frac{1}{129}$. Observing that $100<129<1000$, we compute $\frac{1000}{129}=7\frac{97}{129}$. So the first nonzero digit to the right of the decimal point of $\frac{1}{129}$ is $\boxed{7}$.