jmc

An integer is divisible by $9$ if and only if the sum of its digits is divisible by $9$. Using this fact, we find that the greatest multiple of $9$ that is less than $-1111$ is $-1116$.

We have $-1111=-1116+5$. That is, $-1111$ is equal to a multiple of $9$ plus $5$. Therefore, $$-1111\equiv \boxed{5}(\mathrm{mod}9).$$To verify this answer, we can check that $9$ divides $(-1111)-5=-1116$; it does, so our answer is correct.