Complexity bound on $RP^{RP}$
Hint: For BPP it is the case that $BPP^{BPP} = BPP$. The idea is that given a machine in $BPP^{BPP}$, we simulate each oracle call by a BPP computation, amplified so that the error probability is very small. Since there are only polynomially many oracle calls, overall the error coming from miscalculating the BPP oracle calls is small, and the resulting machine still has bounded error. (For a more complete proof, use your web search skills.) Try to emulate this line of reasoning for RP.