What are some examples of cohomology theories without a corresponding classifying space?

Question

The general nonsense of cohomology theories is that each one "should" be presented by a classifying space, so that maps into this space give the cohomology (before passing to connected components). For cohomology with coefficient group $G$, this is the classifying space $\mathbf{B}G$ (Edit. The representing object here should be the Eilenberg-MacLane spectrum for the group). Cobordism is represented by the universal Thom spectrum of the (stable?) orthogonal group, $MO$. Topological K-theory is represented by a categorification of the integers, generally the smooth stack $\mathbf{Vect}$. However useful this concept is, the $n$Lab consistently says that "most" cohomology theories have a corresponding classifying object. There must be some examples, then, of theories without. What are they, and what prevents them from having classifying spaces?