An equation that generates a beautiful or unique shape for motivating students in mathematics
I'd like to mention Spirographs.
The formulas are actually rather simple, but I'm afraid that my Latex-foo is not sufficient to reproduce them here adequately. So I'll just refer to the Wikipedia page, and some example images (also from Wikipedia):
Fractals are always a good source of pictures. It's not too hard to explain the concept behind a fractal, and then students can enjoy the pretty depictions. Some of them are also easy for students to play with themselves --- for the Koch snowflake, the dragon curve, or the Sierpinski gasket, you don't have to know any complex function theory. Fractals can also lead to neat discussions of "infinity."
Edit: I should have read the question more carefully! Equations. Let me try to salvage my Googling of pretty pictures ...
Often fractals arise from the iterated application of a single function (Julia sets in $\mathbb{C}$ from $z^2 + c$ as the mother of all examples), so they correspond to solution sets of an equation with infinitely nested expressions. You could also write down the procedure for generating the Koch snowflake or the dragon curve as an equation. (Formally, the former is called "snowflaking a metric", but the notation and concepts are probably a bit above your audience.) These also help make the point that, from one perspective, functions are procedural.
Polynomial curves of the form $\displaystyle\sum_{k=0}^na_k\cdot x^{2k}\cdot y^{2(n-k)}=r^{2n}$, with $a_k=a_{n-k}$ . This is for the case
$n=4$ and $r=2$, with $a_0=a_4=0.1$, $a_1=a_3=4$, and $a_2=-7$. By modifying the parameters,
wildly different shapes can be formed.
More star-shaped graphics, determined by plotting the polar equation $r(t)=|\cos(nt)|^{\sin(2nt)}$
for $2n$ in between $1$ and $8$, and $t\in(0,2\pi)$.
