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.
I doubt that there is established notation for this construction. If you need it often in something you are writing, define one for yourself - perhaps one of $$ F(A,c,B) $$ or $$ (A,B)^n(c). $$ or $$ A^n(c;B) $$ depending on how much you want to stress each of $A$, $B$ and $c$.