In the years 1871-1884 Georg Cantor invented the theory of infinite sets. In the process Cantor constructed a set that is self-similar at all scales. Magnifying a portion of the set reveals a piece that looks like the entire set itself.
To construct this set, take a line and remove the middle third. There are two line segments left. Take the remaining two pieces and remove their middle thirds. Repeat this process an infinite number of times. The resulting collection of points is called a "Cantor" set.
The Cantor set is an unusual object. The deletion process produces an infinite set of points. On the one hand, the points are more numerous---more "infinite"---than the integers, since you can't count them. On the other hand, the Cantor set is not a continuum of points like the original whole line. The Cantor set is somewhere in between. The points in the Cantor set are quite close together. Pick any point and you can find another point that is arbitrarily close to it.
At the time Cantor discovered these "pathological" sets---to repeat a phrase used at the time---it was widely believed that they were the purest form of mathematical invention. Never would they be seen in the natural world.
Today, we know that many, many natural processes produce such self-similar objects. Cantor sets and a much wider range of self-similar structures are now called "fractals".