Complexity for nested #P?
The result is in #P.  Suppose there are 50 states, and each state has 10 cities.  Then how many cities are there in total?

Now suppose we have a polytime verifier for states: $S(x)$ outputs 1 iff $x$ is a state.  We also have a polytime verifier for cities: $C(x,y)$ outputs 1 iff $y$ is a city in state $x$.  Then counting the number of states is in #P: it's the number of $x$ such that $S(x)=1$.  If you think a bit, you can see that counting the number of cities is also in #P: it's the number of pairs $(x,y)$ such that $S(x)=1 \land C(x,y)=1$.  Can you come up with a polytime verifier $V$ such that $V(x,y)=1$ iff $x$ is a state and $y$ is a cities in that state?  I bet you can.  Consequently, counting the number of such pairs (i.e., counting the number of cities) is in #P.

Now apply the same idea to your problem, and you should be able to figure out the answer.