The [monster vertex algebra][1].

It is (to date) the central object in [monstrous moonshine][2], since its character is the $q$-expansion of the modular $J$-function, its automorphism group is the monster simple group, and the graded trace of any element of the monster is the $q$-expansion of a genus zero modular function.  The construction of this structure (by Frenkel, Lepowsky, and Meurman) involves ascending a hierarchy of objects that are by themselves quite intricate and beautiful.

1. One begins with the [extended binary Golay code][3] of length 24.  Up to symmetries, it is the unique copy of $\mathbb{F}_2^{12}$ in $\mathbb{F}_2^{24}$, for which any five basis vectors are contained in a unique codeword (i.e., it forms a Steiner $(5,8,24)$ system).  The codewords are separated by Hamming distance at least 8, so even if 3 bits in a code word are changed the error can be corrected.   The automorphism group of the Golay code is the sporadic simple group $M_{24}$ of order 244823040.

2.  Using the Golay code to produce coordinates of generators, one constructs the [Leech lattice][4] $\Lambda$, which is a rather densely packed copy of $\mathbb{Z}^{24}$ in $\mathbb{R}^{24}$.  One can also make the Leech lattice as a subquotient of the even unimodular lattice $I\!I_{25,1}$, which has its own exceptional properties.  Peter Shor mentioned the Leech lattice in another answer, so I'll just note that its automorphism group is a double cover of Conway's sporadic simple group $Co_1$, which has order 4157776806543360000.

3. For any positive definite even lattice $L$, there is a canonical construction of a [vertex operator algebra][5] graded by that lattice, called the lattice vertex algebra $V_L$.  I think physicists say that it is the algebra of chiral symmetries of a conformal field theory describing a bosonic string propagating in the torus $L \otimes \mathbb{R}/L$ (but I may have mixed up the words).  It has an action of the holomorph of the algebraic torus $L \otimes \mathbb{C}^\times$.

4. The "-1" automorphism of the Leech lattice induces an automorphism $\theta$ of $V_\Lambda$, and there is a unique irreducible $\theta$-twisted module $V_\Lambda(\theta)$ that inherits an action of the centralizer $2^{1+24}.Co_1$ of $\theta$ in the automorphism group of $V_\Lambda$.  The monster vertex algebra is formed by taking the direct sum of fixed points: $V^\natural = (V_\Lambda)^\theta \oplus (V_\Lambda(\theta))^\theta$.

Apparently, the hard part was proving that the monster acts on $V^\natural$ by automorphisms.

There are some additional conjectural reasons for considering it beautiful:

1. In the same paper where it was constructed, it was conjectured to be the unique vertex operator algebra with central charge 24, character equal to the modular $J$ function, and representation category equivalent to $Vect$.  (Naturally, this does not account for higher structure like twisted modules.)

2. [Witten suggested][6] that it is dual to pure 3-dimensional quantum gravity with minimal cosmological constant by AdS/CFT correspondence.


  [1]: http://en.wikipedia.org/wiki/Monster_vertex_algebra
  [2]: http://en.wikipedia.org/wiki/Monstrous_moonshine
  [3]: http://en.wikipedia.org/wiki/Binary_Golay_code
  [4]: http://en.wikipedia.org/wiki/Leech_lattice
  [5]: http://en.wikipedia.org/wiki/Vertex_operator_algebra
  [6]: http://arxiv.org/abs/0706.3359