The author analyzes the notions of power boundedness and mean ergodicity for composition operators \(C_\phi\) acting on weighted Bergman spaces of infinite order \(H^\infty_v\). Recall that a bounded operator \(T:X\to X\) is called power-bounded whenever \(\sup_{n\in \mathbb N} \|T^n\|0\) and presents a number of results and examples related to the question when composition operators on weighted Bergman spaces have the above mentioned properties.