The Mandelbrot set is a famous fractal set. The inside of the Mandelbrot set is given by all the parameter values for which the corresponding Julia set (another famous family of fractal sets) is connected. Like all fractal sets, the Mandelbrot set has a lot of structure at increasingly fine scales. This picture shows a zoom-in by 10^12 of a tiny part of the Mandelbrot set, where we see a little imitation of the set again (in black).
Figure credit: Glenn Elert.
URL for this and other pictures of blowups of the Mandelbrot set with coarser scales, or for other pictures of Julia sets: http://hypertextbook.com/chaos/eyecandy