What is the product of this by telescopic method?
The terms of the product are $(1+1/2)(1+1/4)(1+1/16)(1+1/256)\cdots$ with each denominator being the square of the previous denominator. Now if you multiply the product with $(1-1/2)$ you see telescoping action:
$(1-1/2)(1+1/2)=1-1/4$
$(1-1/4)(1+1/4)=1-1/16$
$(1-1/16)(1+1/16)=1-1/256$
Do you see the pattern developing?
$$1+ {\frac{1}{2^{2^k}}}=\cfrac{1- {\cfrac{1}{2^{2^{k+1}}}}}{1- {\cfrac{1}{2^{2^k}}}}=\frac{u_{k+1}}{u_k}$$ hence
$$\prod_{k=0}^{\infty} \biggl(1+ {\frac{1}{2^{2^k}}}\biggr)=\frac{1}{u_0}=2$$