Fractals are found everywhere in nature. This is due to the fact that a fractal can be used to describe just about anything. (this is also why fractals are used for image compression) To find an example of a fractal in nature, look anywhere. One example is a leaf. The branching of the veins creates self-similarity and independence of scale. These two things qualify the leaf as a fractal.
Please note that natural objects are not really fractals, but are actually fractal-like. This is because natural processes act over a finite range of scales. In other words, although natural objects can be zoomed in to infinity, they won't look the same at different zooms. For example, the fractal branching of leaf veins ends at the cellular level and new patterns occur.
The Golden Spiral is a mystical shape that is an absolute in both abstract mathematics and chaotic nature. It was first discovered by Phythagoras, a failed Greek messiah and mathematical cult leader in the 5th century B.C.
The spiral is derived via the golden rectangle, a unique rectangle which has the golden ratio. When squared, it leaves a smaller rectangle behind, which has the same golden ratio as the previous rectangle. The squaring can continue indefinitely with the same result. No other rectangle has this trait.
When you connect a curve through the corners of these concentric rectangles, you have formed the golden spiral. The Phythagoreans loved this shape for they found it everywhere in nature: the Nautilus Shell, Ram's horns, milk in coffee, the face of a Sunflower, your fingerprints, our DNA, and the shape of the Milky Way.
"My new Hypothesis: If we're built from Spirals while living in a giant Spiral, then is it possible that everything we put our hands to is infused with the Spiral?"
-- Max Cohen in the motion picture Pi