Throwing a fair dice until most recent roll is smaller than previous one

I roll a fair dice with $n>1$ sides until most recent roll is smaller than previous one. Let $E_n$ be the expected number of dice rolls. Do we have $lim_{ntoinfty} E_n < infty$? If not, what about $lim_{ntoinfty} E_n/n$ and $lim_{ntoinfty} E_n/log(n)$?