For a topological monoid $M$, the classifying space $BM$ is at least a pointed topological space as far as I know.
From where to where is the construction $B$ a functor actually? Can I plug in an $A_\infty$ space $M$ or even a $H$-space $M$? What do I get in those cases? In particular: Can I apply the $B$ costruction again to get $BBM$?
One can form $BM$ for any $A_\infty$-space and it's more or less the delooping ($A_\infty$-structure on a connected $H$-space $M$ is more or less the same thing as an equivalence $M\cong \Omega X$ for some $X$; the proof is more or less that $M\cong\Omega BM$).
(AFAIR this can be generalized further but then there is a question of what properties do you want from $BM$.)
If $M$ is abelian, $BM$ is again a monoid which is abelian enough so the construction can be iterated (basically one gets $B^nM=M[S^n]$ — i.e. the configuration space of points on $n$-sphere with labels in $M$). In particular, if $G$ is a discrete abelian group, one can define $B^nG$ and it has homotopy type of $K(G,n)$ (and so we get a very explicit description of $K(G,n)$ — $G[S^n]$).
On the other hand, if $G$ is a non-abelian group, $BG$ is not (in general) even an $H$-space (say, any surface is a $BG$ but if $g>1$ it's not an $H$-space).