What do we know about the distribution of Mersenne primes?

Mersenne primes are primes of the form $M_n = 2^n - 1$. I'm wondering how far apart successive Mersenne primes can be. For example, is $M_{n+1} \le O((M_n)^e)$? Or, is $M_{n+1}$ always less than some power of $M_n$? If not, how close together can successive Mesenne primes be in the worst case?

It is not even known whether there are infinitely many Mersenne primes! There are guesses only, based on probabilistic assumptions for which there is no proof.

For a brief survey of some conjectural answers about the distribution of Mersenne primes, please see Wikipedia article on Mersenne Conjectures.

Visualization and discussion of distribution of Mersenne primes is maintained by Chris K. Caldwell on his "Where is the next Mersenne prime?" FAQ page.

Particularly, it discusses the Lenstra–Pomerance–Wagstaff conjecture:

There are asymptotically $$e^\gamma\log_2 \log{x} \approx 1.78\log_2\log{x}$$ Mersenne primes less than $$x$$.

where $$\gamma$$ is the Euler's constant.

(Interestingly, the formula predicts 45.9 Mersenne primes smaller than $$2^{82,589,933}-1$$, though this is actually 51st known Mersenne prime. So those primes currently appear more frequently than expected.)

The page also discusses the gaps between Mersenne primes.