We have \begin{align*} \root 3 \of {x \root 3 \of {x \root 3 \of {x\sqrt{x}}}} &= (x(x(x\cdot x^{\frac{1}{2}})^{\frac{1}{3}})^{\frac{1}{3}})^{\frac{1}{3}} \\ &= (x(x(x^{\frac{3}{2}})^{\frac{1}{3}})^{\frac{1}{3}})^{\frac{1}{3}} \\ &= (x(x \cdot x^{\frac{1}{2}})^{\frac{1}{3}})^{\frac{1}{3}}\\ &= (x(x^{\frac{3}{2}})^{\frac{1}{3}})^{\frac{1}{3}} = (x\cdot x^{\frac{1}{2}})^{\frac{1}{3}} = (x^{\frac{3}{2}})^{\frac{1}{3}} = x^{\frac{1}{2}}=\boxed{\sqrt{x}}.\end{align*}