The part about fractals that I think is cool is that not only does the simple equation yield complexity, but the complexity is both infinitely varied and repetitive at the same time. If you look at that picture of the Mandlebrot set and then go zooming down into "seahorse valley" between the two big circular portions, as you zoom and zoom, you will eventually find a thing that looks (almost) exactly like the original set -- with its own seahorse valley and everything. Every time I do that it blows me away. It is like a very tangible way to have a "holy sh*t" experience about the nature of infinity.

FWIW,
Jim