\sqrt{2{\sqrt{2\sqrt{2}}}}=2^{\frac{1}{2}}\cdot (2^{\frac{1}{2}})^{\frac{1}{2}}\cdot ((2^{\frac{1}{2}})^{\frac{1}{2}})^{\frac{1}{2}}=2^{\frac{1}{2}}\cdot 2^{\frac{1}{4}}\cdot 2^{\frac{1}{8}}=\\=2^{\frac{4+2+1}{8}}=2^{\frac{7}{8}}=\sqrt[8]{2^7}=\sqrt[8]{128}