If $f\left(x\right)$ is an odd polynomial function, then $f\left(f\left(x\right)\right)$ is  

Given that the function $f\left(x\right)$ is odd function 

By the definition of odd function , $f\left(-x\right)=-f\left(x\right)$

Consider $f\left(f\left(-x\right)\right)=f\left(-f\left(x\right)\right)=-f\left(\left(f\left(x\right)\right)\right)$

therefore, the function $f\left(\left(f\left(x\right)\right)\right)$ is an odd function