jmc

For the contents of the innermost square root to be nonnegative, we must have $x\ge 0$. To satisfy the middle square root, we must have $$5-\sqrt{x}\ge 0$$ $$\Rightarrow 25\ge x.$$ Finally, the outermost square root requires $$3-\sqrt{5-\sqrt{x}}\ge 0$$ or $$9\ge 5-\sqrt{x}$$ $$\Rightarrow \sqrt{x}\ge -4,$$ which is always true. Combining our inequalities, we get $$0\le x\le 25,$$ or $x\in \boxed{[0,25]}$ in interval notation.