$\sqrt{2+\sqrt{3}}\ast \sqrt{2+\sqrt{2+\sqrt{3}}}\ast \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\ast \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}=\sqrt{2+\sqrt{3}}\ast \sqrt{2+\sqrt{2+\sqrt{3}}}\ast \sqrt{2^2-(\sqrt{2+\sqrt{2+\sqrt{3}}}{)}^2}=\sqrt{2+\sqrt{3}}\ast \sqrt{2+\sqrt{2+\sqrt{3}}}\ast \sqrt{4-(2+\sqrt{2+\sqrt{3}})}=\sqrt{2+\sqrt{3}}\ast \sqrt{2+\sqrt{2+\sqrt{3}}}\ast \sqrt{4-2-\sqrt{2+\sqrt{3}}}=\sqrt{2+\sqrt{3}}\ast \sqrt{2+\sqrt{2+\sqrt{3}}}\ast \sqrt{2-\sqrt{2+\sqrt{3}}}=\sqrt{2+\sqrt{3}}\ast \sqrt{(2+\sqrt{2+\sqrt{3}})(2-\sqrt{2+\sqrt{3}})}=\sqrt{2+\sqrt{3}}\ast \sqrt{2^2-(2+\sqrt{3})}=\sqrt{2+\sqrt{3}}\ast \sqrt{4-2-\sqrt{3}}=\sqrt{2+\sqrt{3}}\ast \sqrt{2-\sqrt{3}}=\sqrt{(2+\sqrt{3})(2-\sqrt{3})}=\sqrt{2^2-(\sqrt{3}{)}^2}=\sqrt{4-3}=\sqrt{1}=1$