$\sqrt{m\sqrt{n\sqrt{p\sqrt{a}}}}=$  

B: $\sqrt[m]{\sqrt[np]{a}}={\left({\left(a\right)}^{\frac{1}{np}}\right)}^{\frac{1}{m}}=a^{\frac{1}{mnp}}=\sqrt[mnp]{a}$

C: $\sqrt[p]{\sqrt[n]{\sqrt[m]{a}}}={\left({\left(a^{\frac{1}{m}}\right)}^{\frac{1}{n}}\right)}^{\frac{1}{p}}=a^{\frac{1}{mnp}}=\sqrt[mnp]{a}$

D: $\sqrt[p]{\sqrt[mn]{a}}={\left(a^{\frac{1}{mn}}\right)}^{\frac{1}{p}}=a^{\frac{1}{mnp}}=\sqrt[mnp]{a}$

So, all are correct