I think that Barnsleys Fern is a really surprising image, that such complex shapes can be encoded in four very simple affine transformations.

If you allow for a larger class of functions (stochastic, $\mathbb{R}^3 \to \mathbb{R}^3$, and introduce a log-density plot and color each point according to orbit history, the possibilities are endless (image created by Silvia C.):

The most common applications of the latter algorithm seems to be producing abstract book covers for books about the universe:

I think that Barnsley's Fern is a really surprising image, that such complex shapes can be encoded in four very simple affine transformations. 

If you allow for a larger class of functions (stochastic, $\mathbb{R}^3 \to \mathbb{R}^3$, and introduce a log-density plot and color each point according to orbit history, the possibilities are endless (image created by Silvia C.): 

The most common applications of the latter algorithm seems to be producing abstract book covers for books about the universe: