In Woronowicz's theory of compact quantum groups, the most well-known example is $SU_q(2)$, for $q$ a real number. Moreover, all the other examples of compact quantum groups, based some Drinfeld--Jimbo dual, that I can find, are for $q$ not a (non-trivial) root of unity. Does anyone know of an example of a compact quantum group at a root of unity?

This is impossible in the usual sense. For example, you can look at the dimension function on the representation category of $SU_q(2)$. The quantum dimension of the half spin representation is $|q + q^{-1}|$, and in general the quantum dimension cannot be smaller than the usual dimension.