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