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 any $n_4$ 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 $100,000$ can be factors, or at most $21$ of the primes below $1,000,000$) and the likelihood of being prime for some $n_4$, based on the required subsequent statistical considerations only, should be much larger than that of a non-iterated Mersennenumber $n_1$ of the same size.
*Appended: for any $n_5$ we have only $3$ primes below $1,000,000$ as possible primefactors, namely $214559$,$429119$ and $858239$*