This isn't intended as a technical question, but one of aesthetic preference. I most enjoy fractal landscapes that seem real and 3D. Zooming through the Mandelbrot set was my first riveting experience, but there are now many examples of plausible sci-fi like and psychedelic visualizations, such as an application from Fractal Lab.
Fly Above The World from Fractal Lab requires a high end computer and browser, but claims to provide provide "open-source application "Fractal Lab," which gives anyone the power to explore 3-D fractal-planetoids for free."
Going into the spaces of a Sierpinsky sponge reminded me of that abandoned Japanese island crammed with decaying multistory buildings (a set in the latest James Bond film). It looked like a surface upon which you might walk,
crevices where you could sit or store your picnic basket.
In case you're wondering, this is also called a Menger sponge.
A particular fractal, called Menger's Sponge, is all about surface appearances. It's a purely theoretical shape that has infinite surface area and no volume whatsoever. And because of that, it doesn't occupy three dimensions. Or two. It manages to exist in fractional dimensions.
Each of those pitted walls is called a Sierpinski Carpet. We are taught to think of things as having one, two, or three dimensions ..., but the Sierpinski Carpet is supposed to straddle the division between a one-dimensional line and a two dimensional plane. Clearly it occupies an area, but the surface is so pitted that it technically doesn't fill the area, so much as scribble a bunch of lines over it. As crazy as it sounds, the Sierpinski Carpet is supposed to have a fractional dimension of 1.89, and the Menger Sponge, which has no real volume, has a fractional dimension of 2.73.
What do you enjoy?
I've seen this photo in many fractal articles and didn't see anything interesting. Now that I know how it is created, it is fascinating. The repetition of a pattern throughout the structure, from very small to large holes, and repeated throughout that creates a sponge effect.
I like her memory of playing with blocks of wood as a child. Our neighbor burned wood ends from the lumber yard, and I loved going over there on delivery day and building whole villages out of the odd shapes and sizes. I never thought of it as she described:
"Almost everyone played with wooden building blocks at some point as a child. That activity might have seemed pointless at the time, but it will now help you understand a particular fractal. Isn't childhood development weird?"
My favorites are the fractals in nature, whether a brassica or a seashell or a tree. Even in the seemingly chaos of nature, there is order.
Here are organic-looking ones, "Strange Fruit" from roddh's photostream at flickr
How about ginger root as a natural fractal, Joan.
More that look like a sci-fi cloud city.
see more at Mandelbulber-3D
In case you missed the reference, the ones I cited as my favorites were made with Fractal Lab. I do not own the software, just found examples of work by others.