Looks fantastic, this is just what I need for recent work. However, the current algorithm is unable to find some denesting result even when one truly exists, e.g. ones related to non-algebraic numbers and transcendental functions, like this below:
$$ \frac{1}{2} \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}} $$
From my test today, the function posted is unable to find the denesting result of the compound radicals above, but there truly exists one:
$$\frac{1}{2} \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}}=\sin \left(\frac{\pi }{96}\right)$$
So I hope you can improve your function, making it able to solve compound radicals related to non-algebraic numbers and transcendental functions.
Despite its imperfection, I am still very impressive about Corey Ziegler and Bill Gosper, Daniel Lichtblau, Swastik Banerjee and their team's excellent work. This is no doubt a good start. Please keep going, making this function more and more powerful!