jmc

First, notice that $729=3^6$. We can begin simplifying from the innermost square root: $$\sqrt{\sqrt[3]{\frac{1}{\sqrt{729}}}}=\sqrt{\sqrt[3]{\frac{1}{27}}}=\sqrt{\frac{1}{3}}=\frac{1}{\sqrt{3}}=\boxed{\frac{\sqrt{3}}{3}}$$