Assume $A$ to be an operator, $B$ a scalar function - the details should not matter too much here. I know that for an element $c$ the repeated application of the operator can be written as $$ A(A(A(c)))=(A\circ A\circ A)(c)=A^3(c). $$
I am looking for a notation of the repeated composition $$ A(A(A(c)\cdot B)\cdot B)\cdot B $$ where $\cdot$ is multiplication.