*Remark: I'm composing my two comments at the OP making them an answer*             

Mersenne-numbers and iterated Mersenne-numbers of the same size have the significant difference, that the iterated ones cannot have "small" prime-factors while for the basic Mersenne numbers there is no such restriction. For instance a two-time iterated Mersenne number cannot have primefactors $3,5,11,...$ and only $7,23,..$ can be primefactors of such a number. So the possible number-of-primefactors for highly iterated Mersenne-numbers is much smaller than a naive expectation based on that for non-iterated Mersenne-numbers of the same size.                   

This looks even more drastical for higher iterates. Let's denote $n_0$ a variable having any positive integer value, $n_1=2^{n_0}−1,$, $n_2=2^{n_1}−1$ and so on. Then $n_4$ can have at most two of the primes below $10,000$ as factors, namely $2879$ and $4703$ (or: at most $5$ of the primes below $100,000$ as factors, or at most $21$ of the primes below $1,000,000$) and the likeliness of being prime of some $n_4$, based on subsequent statistical considerations only, should be much larger than that of a non-iterated Mersennenumber $n_1$ of the same size.              

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*Appended: for $n_5$ we have only $3$ primes below $1,000,000$ as possible primefactors, namely $214559$,$429119$ and $858239$*