What's a cohomology that's not defined from a cochain complex?

Wikipedia is talking about generalized cohomology theories. They satisfy a bunch of axioms formally similar to what e.g. singular cohomology satisfies. But the so-called extraordinary cohomology theories (K-theory, cohomotopy, cobordism and so on) do not come from chain complexes. Indeed, by a theorem of Burdick--Conner--Floyd, in some sense if they did "come from chain complexes", then they would be given by a product of ordinary cohomology theories.

R. O. Burdick, P. E. Conner, and E. E. Floyd. “Chain theories and their derived homology”. In: Proc. Amer. Math. Soc. 19 (1968), pp. 1115–1118. issn: 0002-9939.

Roughly speaking, the theorem is that if you have a generalized cohomology theory $h$ given by the cohomology of a (functorial on pairs) cochain complex, and such that the long exact sequence comes from the long exact sequence of that cochain complex, then $$h^n(X,A) = \sum_{i+j=n} H^i(X,A; h^j(\text{point}))$$ is really given by singular cohomology with various coefficients.

But the emphasis here is on "cohomology theory". In a different setting (homological algebra), "cohomology" always means "cohomology of a cochain complex". Honestly, the Wikipedia article should probably be renamed to something like "Cohomology theories" or "Cohomology (topology)" because that's what it's really about.

There are various generalized cohomology theories which are typically not described in terms of cochain complexes, perhaps the simplest example being topological K-theory. There's a precise technical sense in which topological K-theory cannot be "described in terms of cochain complexes," but it's not easy to spell out.

Our topology professor, Prof. Michael Weiss, defined in his lectures a homology theory for topological spaces (which I think he mentioned to be equivalent to Čech cohomology) by the means of something which he calls mapping cycles. These are elements of the sheafification of the free abelian group of continuous maps between any two topological spaces. Homology and cohomology are then defined as the groups of mapping cycles from and to spheres respectively, modulo homotopy and “constant” mapping cycles. There are no chain complexes involved.

Accidentally, there’s also a question here on math stackexchange which gives a short overview of the construction, namely here. The questioner also refers to the same lectures I refer to.

The lecture notes (in English) of his topology lectures in 2013/2014 and 2014/2015 at the University of Münster found on his web page give this definition of homology in chapter 5, section 3. Mapping cycles are introduced in chapter 4. Weiss calls it homology without simplices in contrast to the tradition of defining homology via simplices (and chain complexes) since Poincaré.

To my knowledge, the construction is due to Weiss himself.