Domain.

Let $f(x) = \sqrt{1-\sqrt{2-\sqrt{3-x}}}$ . For $f(x)$ to be real,

$\begin{cases} \sqrt{2-\sqrt{3-x}} \le 1 & \implies 2 - \sqrt{3-x} \le 1^2 & \implies \sqrt{3-x} \ge 1 & \implies x \le 2 \\ \sqrt{3-x} \le 2 & \implies 3-x \le 4 & \implies x \ge - 1 \end{cases}$

Therefore, $x \in [-1,2]$ .