Q:

When few simple conditions yield a unique intricate structure

If people were asked to do a brainstorming related to the title, everyone would probably come up with dozens of examples. Those could include things as different as  

the Mandelbrot set, Julia sets and all other sorts of fractals
the j-invariant and other special functions
the sporadic simple groups (not unique, but almost, in the sense that there are only a small number of them)
lots of other well known examples...

Obviously there would be no big use suggesting anything that broad. 
To come up with a reasonably short list, I would like to exclude  

structures that are largely known (your judgment!)  
structures that are mainly defined by some extremality condition (e.g. cage graphs).

Examples for the sort of surprising discoveries I would appreciate to see mentioned here are things like Laver tables (which actually inspired this question) or the Feigenbaum Function, unique solution of the Feigenbaum-Cvitanović functional equation. In this vein:

Which sophisticated structures with easy defining conditions (and whose existence is possibly not straightforward) from your favorite domain would you like to share?

A:

The Leech lattice $\Lambda_{24}$ answers your question with quite a large multiplicity.

The unique $24$-dimensional laminated lattice, defined by $\Lambda_0$ being the one-point lattice and $\Lambda_{n+1}$ being a maximum-density lattice packing obtained from adjoining an additional vector to $\Lambda_n$.
The lowest-dimension even self-dual unimodular lattice with no roots (vectors of norm $2$).
The lattice of vectors in the Lorentzian lattice $II_{25,1}$ perpendicular to the light-like Weyl vector $(0, 1, 2, 3, \dots, 24; 70)$.
The unique densest lattice packing of balls in $24$-dimensional space.
The lowest-dimension even self-dual unimodular lattice that is not directly congruent to its enantiomer.

There are lots of other characterisations, for example based on the relationship between its theta series and the Ramanujan tau function.

