I'd just like to point out that there is a monad on $Top$, (which in the homotopy category looks rather dull,) assigning to each space $X$ its cone $CX$, the mapping cylinder of $X\to * $.  The unit map is the inclusion of $X \to CX$, and the composition may as well be the map $ [[x,s],t]\mapsto [x,s+t-ts].$  (This ugly formula is just a natural obfuscation of the heuristic description of $CX$ as the union of convex combinations of points $x\in X$ and the new point $*$.)  Another way to think of it is that $CX$ is the underlying space of the free contraction of $X$.

The topological (realization of the) simplex category is just the orbit of a one-point space $\star$ under this monad, together with the maps derived from the monad and the maps $C\star\to \star$ and $\star\to *\to C\star$.  I think this gets at what [Grigory M](http://mathoverflow.net/users/1556/grigory-m) means [above](http://mathoverflow.net/questions/28380/a-canonical-and-categorical-construction-for-geometric-realization/36692#36692) by "most natural" contractible set on $n$ points.  Somewhere in this nonsense I should say "Bar construction", but I can't remember precisely where.