jmc

We can write $$\begin{array}{rl}{2}^{\frac{1}{2}}\cdot 4^{\frac{1}{4}}\cdot 8^{\frac{1}{8}}\cdot 16^{\frac{1}{16}}\cdots & =2^{\frac{1}{2}}\cdot 2^{2\cdot \frac{1}{4}}\cdot 2^{3\cdot \frac{1}{8}}\cdot 2^{4\cdot \frac{1}{16}}\cdots \\ & =2^{\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\cdots }.\end{array}$$Let $$S=\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\cdots .$$Then $$2S=1+\frac{2}{2}+\frac{3}{4}+\frac{4}{8}+\cdots .$$Subtracting these equations, we get $$S=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots =\frac{1}{1-1/2}=2,$$so $$2^{\frac{1}{2}}\cdot 4^{\frac{1}{4}}\cdot 8^{\frac{1}{8}}\cdot 16^{\frac{1}{16}}\cdots =2^S=2^2=\boxed{4}.$$