A unimodal map is function $f:[0,1]\rightarrow [0,1]$ such that there exist $c\in (0,1)$ such that $f$ is strictly increasing on $[0,c)$ and strictly decreasing on $(c,1]$.
A unimodal map f is renormalizable if there is a proper subinterval $J$ of $[0,1]$ and an integer $n>1$ such that $f^n|_J$ is itself (topologically conjugate to) a unimodal map.
Of course we can ask ourselves whether we can renormalize the renormalized function again. My question is:
Can we determine the maximal number of renormalizations we can perform for a given unimodal function?
In case this is too difficult, is it known for the logistic family $f_\lambda(x)= \lambda x (1-x)$?
Added: I forgot to mention that $f^n(J)\subseteq J$.