Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{\log^n(n)}$$ While proving the convergence of this series is trivial (e.g., using the root test), finding a closed-form expression for the value to which it converges seems to be hard. The summation converges to $ \displaystyle\approx 3.24261$. After various searches on Google and other sites, unfortunately I did not find any useful information.
What does this series converge to? In particular, does a closed-form expression exist for this limit? I would also be interested in knowing whether there are contexts where this series arises.