An example where this does work quite nicely has to do with Bring radicals or "ultraradicals [1]. One of the most important results from Galois theory is that the quintic equation has no solution using standard radicals. But the introduction of "Bring radicals" allows quintic equations to be formally solved. As far as I'm aware though, Bring radicals only work for quintic equations in general and don't work for 6th order or higher polynomials, so your bang for the buck is a somewhat limited.

[1]: https://en.wikipedia.org/wiki/Bring_radical