Using inclusion-exclusion, we can give the following explicit formula:

$$\mathrm{lcm}(2^1-1,2^2-1,\dots,2^n-1) = \prod_{k=1}^n (2^k-1)^{M(\lfloor n/k\rfloor)},$$
where $M(\cdot)$ is [Mertens function](https://en.wikipedia.org/wiki/Mertens_function).

Ignoring "$-1$" in the factors, the last expression can be approximated by
$$2^{\sum_{k=1} kM(\lfloor n/k\rfloor)} = 2^{\Phi(n)} \approx 2^{\frac3{\pi^2}n^2},$$
giving the same estimate as in the other answers, where $\Phi(\cdot)$ is the [totient summatory function](https://en.wikipedia.org/wiki/Totient_summatory_function). Perhaps, this estimate can be made rigorous, but I'll leave it here just as an idea.