Individual mathematical objects whose study amounts to a (sub)discipline? 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) $SL_2({\mathbb R})$;
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!
 A: The moduli spaces of curves, $\overline{\mathcal{M}}_{g,n}$.
A: The braid group. 
The Monster group. 
The Steenrod algebra. 
The representation ring of the symmetric group.  
A: The homotopy groups of spheres, $\pi_k(S^n)$.
A: $\pi{}{}{}{}$ ${}{}{}{}$
A: The Korteweg–de Vries equation. For almost 90 years it was seen as just another non-linear equation stemming from fluid dynamics. Everything has changed after people discovered the world of solitons.    
A: $E_8$
separable Hilbert space
maybe, Thompson's group $F$
A: The Fermat equation $x^n+y^n=z^n$ is a candidate I guess.
A: Hopf fibration, Icosahedron, Henon map, Hilbert (space filling) curve, Conic sections
A: $SL_2\mathbb R$ and its evil universal covering.
A: Conway's Game of Life  in 2-dimensions, as my exemplar instance in the class of (what used to be my overly general answer of...) Automata: deterministic finite state machines and nondeterministic and probabilistic automata and the theory behind them leading to things like acceptors of regular languages and the concepts of simulation, computational equivalence and computability as in Turing machines and "Turing equivalent", and the concept of "power of computing", computational complexity and complexity classes, bisimulation (and the equivalent computing power of single-tape vs. multi-tape and other classes of Turing machines, and the equivalent computing power of systems which can simulate other systems). 
A: The Erdos-Renyi random graph model $G(n,p)$ - a single, concrete model that more or less created the field of random graph theory and is still studied. 
A: $C[0,1]$.  Since every separable metric space embeds isometrically into $C[0,1]$ and every separable Banach space embeds isometrically isomorphically into $C[0,1]$, the study of $C[0,1]$ includes the study of the geometry of separable metric spaces as well as 90% of  Banach space theory.
A: Godel's constructible universe L.
A: The free group factor(s) -- not just because of the infamous free group factor problem, but also because, IIRC, $VN(\mathbb F_2)$ and relatives appeared very early on when von Neumann et al. were laying out the theory and looking for examples to demonstrate its richness.
A: The hyperbolic space.
A: Knots.  Quandles and Racks.
