Short version. What upper bounds are known, for the number of divisors of Mersenne numbers?
Long version.
Studying the structure of the factors of $M_n = 2^n - 1$ appears to be an active and difficult research problem. Of course, famously, those values of $M_n$ which are prime (i.e., the Mersenne primes) are ones often considered in searches for large prime numbers, and it appears to be folklore that Mersenne numbers more generally have relatively few prime factors (as also observed in the integer sequence A046051). Some of this is due to the properties that a prime divisor of a Mersenne number must have (see this answer to a related question on Math SE), which means that on just statistical grounds that the prime factors of a given Mersenne number should be rare.
But despite much searching, I haven't found any results on upper bounds on the number of factors — prime or otherwise — of a Mersenne number. Much of the research relating to this seems to me indirect, such as analysing $\sum_{k \mid M_n} \tfrac{1}{k}$, which seems tangentially related but without a useable relationship: see Ref. [1, p.460]. The closest to any overt statements that I could find about it in the literature, are of the sort from Pomerance [2]:
It is to be remarked [...] that the situations for $d(m)$, $\omega(m)$ [for $m$ a Mersenne number] are much harder and are far from resolution. Here $d$, $\omega$ respectively count the number of natural divisors and the number of distinct prime divisors.
and note particularly the remark on the final page,
note that [...] $2^p - 1$ has fewer than $p$ prime factors
which is pretty unhelpful, considering that absolutely all integers $m > 2$ have fewer than $\log_2(m)$ prime factors [3]. For there not to be any easily found remarks on a better bound for Mersenne numbers, is very surprising to me.
It seems to me that one can probably obtain some non-trivial bounds on the number of divisors of Mersenne numbers. For instance, because any prime divisor of $M_n$ must be a number of the form ${2mn + 1}$ for an integer $m > 0$, we can obtain an upper bound on the number of factors of any $M_n$ which happens to be square-free, just by an argument considering products of consecutive factors ${2mn + 1}$ for $1 \leqslant m \leqslant k$, and applying an upper bound involving $k!$. A back-of-the-envelope calculation suggests to me a bound of $\omega(M_n) = \tilde O(\sqrt n)$, so that $d(M_n) = 2^{\tilde O(\sqrt n)}$. I also suspect that the squarefree Mersenne numbers are, asymptotically speaking, the ones with the most factors: this might not even be very difficult to prove.
Perhaps I'm making some mistakes here, but I would have expected there to be some bound of the form $2^{cn^\delta}$ for some $c > 1$ and $\delta < 1$, for the number of divisors of a Mersenne number. But I can't find any results; neither from research papers nor lecture notes, unless it is a special case of a more general sort bound that I haven't noticed. What results are known in this direction?
References
Luca, Shparlinski. Arithmetic functions with linear recurrence sequences. Journal of Number Theory 125 (pp. 459–272), 2007.
Pomerance. On primitive divisors of Mersenne numbers. Acta Arith. 46 (pp.355–367), 1986.
This can be shown by a simple counting argument, particularly as there are no proper divisors of $m$ which are greater than $m/2$. More generally, it seems to me that using the divisor bound, one can show that the number of prime divisors of integers $m$ in general scales as $O(\log(m)/\log \log (m))$; and I think that a similar, concrete bound on the number of prime factors of a Mersenne number $m = M_n$ can be shown quite easily, again by taking into account the restrictions on the prime factors that they can have. So a better bound is readily available, but even that bound is not particularly special to Mersenne numbers.
