Not-so-amazing nested radicals

The expression above is equal to $7^{\frac{1}{2}}\cdot7^{\frac{2}{4}}\cdot7^{\frac{3}{8}}\cdot7^{\frac{4}{16}}\cdots$

= $7^{\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\cdots}$

By the formula for arithmetic-geometric progression, the value of $\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\cdots$ is equal to $\frac{\frac{1}{2}}{1-\frac{1}{2}}+\frac{\frac{1}{2}\cdot\frac{1}{2}}{(1-\frac{1}{2})^2}=2$

So, our product is equal to $7^2 = \boxed{49}$