ACC0 and the majority gate
It does not reach a constant ceiling, but the modulo gate cannot depend on the input's length, e.g. $\bmod n/2$ gates are not allowed. You must decide on the constants while constructing the circuit, independently of the input/input size. See the definition in Arora and Barak's book, where they first define $ACC0(m_1,...,m_k)$ which corresponds to circuits with $\bmod m_1,...,\bmod m_k$ gates, and then define $ACC0$ as the union of $ACC0(m_1,...,m_k)$ over all integer $k$-tuples.