Is $P^{\#P}=(P^{\#P})^{\#P}$ ? Intuitively, this equation holds because given the second #P oracle can be omitted since we can always use the first one.

More generally, say O is an oracle, is $P^{O}= (P^{O})^{O}$?
You should probably take a look at Is $(NP^{NP})^{NP} = NP^{(NP^{NP})}$? and Are Oracles Associative?. I don't know any canonical definition of $(A^B)^C$ hence I've got the same problem as Tsuyoshi Ito.

If only the $P$ machine has access to oracle $O$, then yes, ${(P^O)}^O=P^O$ by definition.

If both the $P$ machine and the oracle $O$ can both query the oracle $O$, then just take $O=EXP$ as a counterexample: $(P^{EXP})^{EXP} = TIME(2^{2^{n^{O(1)}}})$ whereas $P^{EXP}=TIME(2^{n^{O(1)}})$.