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 R that literally has no components-it contains no open intervals of 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.