Challenge · Superchallenge problem

Compute the exact value of the infinite product $\left(1 + \dfrac{1}{2}\right)\left(1 + \dfrac{1}{4}\right)\left(1 + \dfrac{1}{16}\right)\left(1 + \dfrac{1}{256}\right)\cdots$, where each factor is $1 + 2^{-2^k}$.