$$
\begin{gathered}
\sqrt{2+\sqrt{3}}, \sqrt{2+\sqrt{2+\sqrt{3}}}, \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \\
\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}
\end{gathered}
$$

Multiply the following nested radicals together:

$$
\begin{gathered}
\sqrt{2+\sqrt{3}}, \sqrt{2+\sqrt{2+\sqrt{3}}}, \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \\
\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}
\end{gathered}
$$
We perform the multiplication in the following order:

$$
\begin{aligned}
& \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}} \cdot \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \cdot \sqrt{2+\sqrt{2+\sqrt{3}}} \cdot \sqrt{2+\sqrt{3}}= \\
& =\sqrt{2-\sqrt{2+\sqrt{3}}} \cdot \sqrt{2+\sqrt{2+\sqrt{3}} \cdot \sqrt{2+\sqrt{3}}}=\sqrt{2-\sqrt{3}} \cdot \sqrt{2+\sqrt{3}}=1
\end{aligned}
$$