The value of $(\sqrt{1+\sqrt{1+\sqrt{1}}})^{4}$ is:
(a) $\sqrt{2}+\sqrt{3}$
(b) $\frac{1}{2}(7+3 \sqrt{5})$
(c) $1+2 \sqrt{3}$
(d) 3
(e) $3+2 \sqrt{2}$
$$
(\sqrt{1+\sqrt{1+\sqrt{1}}})^{4}=(1+\sqrt{2})^{2}=1+2 \sqrt{2}+2=3+2 \sqrt{2}
$$

The correct option is (e).