It has puzzled me for a long time that **the Feigenbaum constant $\delta$ and reduction parameter $\alpha$ do not seem to be related to other constants** (that is, numerically), even not to each other. In fact I have never seen them expressed as an integral, any kind of series or product, a nested expression, ... The only thing I have found on the internet is this algorithmic approach which further links to this, but it seems rather like an "a posteriori" method, being interested more in the algorithm than in the nature of $\delta$. Anyway, I would *not* expect the prominent occurrence of the number $163$ in the algorithm to have a deeper meaning.

On the other hand, I wouldn't be *too* surprised to see $\delta$ written as a continued fraction with "defineable" terms, given the fact that it can be defined by
$\delta=\lim\limits_{n\to\infty}\dfrac{\mu_n-\mu_{n-1}}{\mu_{n+1}-\mu_n}$, where the $\mu_n$ are the bifurcation points of an iterated map. But I have never seen one either. Any leads?