Let $u_1,u_2,u_3,\dots$ be a permutation of the integers greater than $2$.  

A unit circle is in a regular $u_1$-gon, which is a regular $u_2$-gon, which is in a regular $u_3$-gon, *ad infinitum*. Each polygon is as small as possible (they do not have to be concentric).

>What permutation $u_1,u_2,u_3,\dots$ minimizes $R$, the radius of the outermost polygon?  
And what is an approximation of that minimum $R$?

Could the optimal permutation just be $3,4,5,\dots$ ?

Based on page 7 of [Tightly Circumscribed Regular Polygons][1], an upper bound for $R$ is $2\times 3.5809=7.1618$ (I multiply by $2$, because in the article the cetral circle has radius $1/2$.) I suspect $R$ could be much lower, maybe around $5$.

Related: [Smallest regular $m$-gon covering a regular $n$-gon][2]


  [1]: https://arxiv.org/pdf/1301.6293.pdf#:~:text=regular%20polygon%20circumscribing%20another%20regular%20polygon%20%28with%20a,drawing%20intermediate%20spacing%20circles%20between%20the%20polygon%20pairs.
  [2]: https://mathoverflow.net/q/385442/494920