Getting to the root of the problem

$\large x = \sqrt{5\sqrt[3]{25\sqrt[4]{125\sqrt[5]{625\sqrt[6]{\cdots}}}}}$

$\large x = x$

$\large x = \sqrt{x^2}$

$\large x = \sqrt{x \cdot x}$

$\large x = \sqrt{x\sqrt[3]{x^3}}$

$\large x = \sqrt{x\sqrt[3]{x^2 \cdot x}}$

$\large x = \sqrt{x\sqrt[3]{x^2\sqrt[4]{x^4}}}$

$\large x = \sqrt{x\sqrt[3]{x^2\sqrt[4]{x^3 \cdot x}}}$

$\large x = \sqrt{x\sqrt[3]{x^2\sqrt[4]{x^3\sqrt[5]{x^5}}}}$

$\large x = \sqrt{x\sqrt[3]{x^2\sqrt[4]{x^3\sqrt[5]{x^4 \cdot x}}}}$

$\large x = \sqrt{x\sqrt[3]{x^2\sqrt[4]{x^3\sqrt[5]{x^4\sqrt[6]{x^6}}}}}$

$\large x = \sqrt{x\sqrt[3]{x^2\sqrt[4]{x^3\sqrt[5]{x^4\sqrt[6]{\cdots}}}}}$

$\large \sqrt{5\sqrt[3]{25\sqrt[4]{125\sqrt[5]{625\sqrt[6]{\cdots}}}}} = x = \sqrt{x\sqrt[3]{x^2\sqrt[4]{x^3\sqrt[5]{x^4\sqrt[6]{\cdots}}}}}$

$\large \sqrt{5\sqrt[3]{5^2\sqrt[4]{5^3\sqrt[5]{5^4\sqrt[6]{\cdots}}}}} = x = \sqrt{x\sqrt[3]{x^2\sqrt[4]{x^3\sqrt[5]{x^4\sqrt[6]{\cdots}}}}}$

From observation we can determine that $\large \boxed{x = 5}$