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!

Very good! Is there another similar function that does similar things when variables (rather than just numbers) are involved?

For example, there is the function

denestSqrt[e_, domain_, x_] := Replace[y /. Solve[Simplify[Reduce[Reduce[y == e && domain, x], y,
   Reals],  domain], y], {{r_} :> r, _ -> e}]

written by @CarlWoll (denestSqrt).

Here is an example:

denestSqrt[Sqrt[(1 - 2 x) (1 - x - 2 x^2) (2 - x + 2 Sqrt[1 - x - 2 x^2])], 1/3 < x <= 1/8 (2 + Sqrt[2]), x]
(* 1 - x - 2 x^2 + Sqrt[-(1 + x) (-1 + 2 x)^3] *)