Ok, I'll throw my hat in the ring: I like the classical Cantor set. 

Not only does it demonstrate the complexity relatively simple subsets of the real line have,  it illustrates an important property of measures on the real line - namely, that measurability has nothing to do with cardinality of the set (i.e. this is an uncountable set with measure zero!)     

It also gives an example of a completely disconnected subset of $\mathbb{R}$ that literally has no components - it contains no open intervals of $\mathbb{R}$ in its power set.

There are many, many more observations one can make about the Cantor set, but I think the obvious ones make my point very nicely. When I teach real analysis, this is an example I think I'll be using a great deal to illustrate properties of the real line.