Fascinating post. I had always assumed that the 3rd most irrational number would be the third metallic mean given by n = (n+ sqrt(n^2+4))/2, and subsequently the fourth metallic mean etc. The metallic means also pack the disks nicely. I have recently had my interest in them sparked after I came across solution to point vortex equilibria involving them.

Do you know what the metallic means are bounded by? Are they as bad as the silver ratio/(1+sqrt(2))?

These most irrational numbers, (9+sqrt(221))/10, (13+sqrt(1517))/26... how interesting that they are not just the simple generalization of the continued fraction for the golden ratio.