Certain mathematical objects have a theory so rich that their study
alone arguably constitutes a distinct (sub)discipline.  My own list
would begin with

1) the absolute Galois group of the rationals;

2) the Mandelbrot set;

3) the Stone-Cech compactification of the integers;

4) the three-dimensional Cremona group;

5) the Riemann $\zeta$-function;

6) the hyperfinite type $II_1$ factor;

7) the set of rational prime numbers; 

8) $SL_2({\Bbb R})$;

9) the 27 lines on a cubic surface;

etc.

I suppose one might add "the real line," "the Euclidean plane," "the axioms
of ZFC," but I'm looking for objects that have emerged out of research and whose
richness itself might carry an element of surprise, rather than objects
purpose-built for their universal or foundational character.

I think a survey of such objects would make a lovely text for an undergraduate
capstone course, so I'm asking for your favorite examples.

My question has a sociological underpinning - there actually exist communities
of mathematicians who would recognize the objects I've listed as central to their
focus.  I'm not allergic to suggestions of objects that *should* enjoy that level
of attention, but for whatever reason, don't yet.

In the same spirit, I recognize that all the objects mentioned belong to broad
categories, and could thus abstractly could be deemed mere examples, and certainly
then studied in a broader context.  But *de facto*, these objects enjoy a distinctive critical level of attention in relative isolation.  For example, each makes an appropriate subject for a monographic treatment.  But please don't hesitate to make a suggestion because your favorite object doesn't have a monograph yet!