In addition to the construction of (generalized) Eilenberg–MacLane spaces mentioned in the comments, there are many other examples.
For example, the Dold–Kan correspondence allows us an easy perspective on cohomology operations on simplicial cochains. If $X$ is a simplicial set and $A$ is an abelian group, then $$\def\C{{\sf C}}\def\N{{\sf N}}\def\Z{{\bf Z}}\def\Hom{\mathop{\sf Hom}}\C^*(X,A)=\Hom(\N\Z[X],A).$$ In this formulation, it may not be entirely obvious why we could have nontrivial cohomology operations $$\def\H{{\sf H}}\H^m(X,A)→\H^n(X,B).$$ Consider now rewriting $\H^m(X,A)$ using the Dold–Kan correspondence: $$\def\U{{\sf U}}\H^m(X,A)=[\N\Z[X],A[m]]=[\Z[X],Γ A[m]]=[X,\U Γ A[m]],$$ where $\U$ denotes the forgetful functor from simplicial abelian groups to simplicial sets. Of course, the term $\def\K{{\sf K}}\U Γ A[m]=\K(A,m)$ is the $m$th Eilenberg–MacLane space of $A$.
In addition to providing us with a very simple proof that singular cohomology is representable by Eilenberg–MacLane spaces (which can be seen as another application), we also see immediately that cohomology operations $$\H^m(X,A)→\H^n(X,B)$$ correspond to homotopy classes of maps $$\K(A,m)→\K(B,n),$$ which themselves can be computed as elements of $\H^n(\K(A,m),B)$. Thus, the cohomology groups of Eilenberg–MacLane spaces compute cohomology operations.