What are some numbers/families of numbers, asides from $\Phi$, $\pi$, and $e$ which have interesting algebraic properties? 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 $e$)
 A: 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 $1, p, p^2, p^3, p^4$.)
A: The Gaussian Integers are neat. They form a ring defined as follows:  
$\mathbb{Z}[i] = \{ a+bi : a,b \in \mathbb{Z}\}$
Here is a link to a paper entitled Algebraic Ordinals which I haven't read but perhaps it will be of interest to you.
https://arxiv.org/abs/0907.0877
