Mandelbrot set, a famous fractal that is a badge of honor for mathematicians. One way of defining the Mandelbrot set is by looking at how complex functions behave under repeated iteration.
This video by Softology is about 3d cellular automata. Visions of Chaos is a professional high end software application for Windows. It is simple enough for people who do not understand the mathematics behind it, but advanced enough for fractal enthusiasts to tweak and customise to their needs.