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 $$2-\sqrt{x}\ge 0\Rightarrow 4\ge x.$$ Finally, the outermost square root requires $$1-\sqrt{2-\sqrt{x}}\ge 0.$$ This gives us $$1\ge 2-\sqrt{x}\Rightarrow x\ge 1.$$ Combining our inequalities, we get $1\le x\le 4$, or $x\in \boxed{[1,4]}$ in interval notation.