Recently I've been learning about the algebraic properties of the golden mean (and in general the metallic means). For those unfamiliar, the golden mean (which I'll denote as $\Phi$) is equivalent to $\frac{1+\sqrt{5}}{2}$, and the $a$th metallic mean (which I'll denote as $\Phi_a$) is defined as $\frac{a+\sqrt{a^2 +4}}{2}$, where $a\in Z^+$. Note that the "golden mean" is simply the $1$st metallic mean, when $a=1$.
One property I've found extremely interesting is the fact that for the golden mean, $\Phi^n = \Phi^{n-1} + \Phi^{n-2}$. In more general terms for the metallic means, $\Phi_a^n = a\,\Phi_a^{n-1} + \Phi_a^{n-2}$. This property is significant because it allows you to define exponentiation of these numbers in terms of their previous powers. One such consequence is that for the golden mean, $\Phi^n = F_n \Phi +F_{n-1}$, with $F_n$ being the Fibonaccie sequence.
My question is, what other numbers or families of numbers have interesting algebraic properties such as these? (Asides from $\pi$ and $e$)
The plastic number (or constant) has much in common with the golden ratio and I would say is one of the more interesting numbers. The plastic number is the only real solution to the equation $p^3=p-1$ and arises in connection withe the Padovan sequence, where it is the limiting ratio of successive terms. Moreover, it figures in a Binet-type expression for said sequence.
In fact, the golden ratio and the plastic number are the only two $morphic$ numbers. (Reference: J. Aarts, R. Fokkink, and G. Kruijtzer, “Morphic Numbers,” $Nieuw \ Arch. Wiskd.$, 5 (2) (2001) 56–58.)
By definition, a real number $p>1$ is a morphic number if there exists natural numbers $k$ and $l$ such that
$$p+1=p^k \ \ \text{and} \ \ p-1=p^{-l}$$
The values of $[k,l]$ for the golden ratio and plastic number are $[2,1]$ and $[3,4]$ , respectively.
In addition, just as the square is the golden rectangle's gnomon, the equilateral triangle is the plastic pentagon's gnomon. There is (in all likelihood) no finitary polygon whose gnomon is a regular polygon other than the golden rectangle and plastic pentagon. (The plastic pentagon has sides $1, p, p^2, p^3, p^4$.)