I am skeptical about the relationship described in the previous post, but it would be beautiful if true. And there has been progress towards making it plausible.
Perhaps the clearest way to classify the available theoretical approaches is how they interpret the doubling cascade. Feigenbaum's constant has the rather abstract meaning, of describing "how quickly" a dynamical system goes from a regime of stasis, to switching between two states, to switching between four states (and so on through powers of two, until chaos is reached), as a control parameter is adjusted. How could that be relevant to the probability that an electron emits a photon?
Angel Garces Doz (who has already appeared many times in this blog) in effect proposes to identify the doubling cascade with the cloud of virtual particles - iterated creation of virtual pairs. He points out that the size of the bulbs budding from the Mandelbrot set also diminishes according to Feigenbaum's constant, and says, let's think of spherical wavefunctions in the virtual cloud in this way. It's a brilliantly vivid intuition.
Meanwhile, I found that Mario Hieb's discovery had already appeared (in a different form) in papers by Vladimir Manasson (2006, 2008). His idea is that there is a prototypical self-organizing system (e.g. think of a soliton), that has 1-state, 2-state, 4-state... forms according to the value of some parameter, as in the doubling cascade. His idea is that the different elementary particles correspond to the different states, and that the levels of the cascade correspond to the fundamental forces. The Feigenbaum ratio, describing how the control parameter changes from one level to the next, maps to the relative strength of the forces!
Finally, I take inspiration from Sergei Gukov, for whom the doubling cascade describes the proliferation of fixed points in a renormalization group flow, as some property of a QFT varies. I am also intrigued by the criticality of the Higgs boson mass in this regard.
I am wondering whether - for example - you could have dissipative "hypermagnetization" (hypercharge magnetization) of the QCD vacuum in the early universe, passing through a doubling cascade of dynamical regimes, and ending in a top quark condensate that breaks electroweak symmetry, leaving only the familiar electromagnetic interaction, with Feigenbaum's constant somehow imprinted on the size of the electromagnetic coupling.