As mentioned before, nature is full of fractal-like stuff. Twigs on trees look like the branches which they grow on, which look like the tree itself. Its the same thing with fern leaves, and so many other living things. Self-similarity is part of this world, so fractals can make pretty good copies of it. Artists have created very realistic looking landscapes composed of just a few fractal equations.
Fractals also have technological applications.
Antennas have always been a tricky subject. Many antenna engineers have been reduced to using trial and error because of the complex nature of electromagnetism. The usual long, thin wire isn't the best way. Antenna arrays, another approach, consist of thousands of small antennas which are either placed randomly or regularly spaced. Fractals provide the perfect mix between randomness and order, and with fewer components.