Complexity classes where $C^C = C$
$\mathrm{BQP}^{\mathrm{BQP}} = \mathrm{BQP}$ has been proved in Strengths and Weaknesses of Quantum Computing Bennett et al. (arXiv).

According to the complexity zoo, $\mathrm{ZBQP}^{\mathrm{ZBQP}} = \mathrm{ZBQP}$.