Foundational uniqueness and representability results on (co)homology theories in algebraic topology frequently make a point of assuming additivity, indicating at least some people think it's worthwhile considering theories not satisfying the wedge axiom despite the inapplicability of familiar results. 


1. **What are some natural examples of such theories (and the failure of additivity)?**
2. What are some stupid examples?



(Co)homology theories not satisfying the wedge axiom
Here's an example. (Rather than calling it a stupid example, I'll call it an example for which I know no applications.)


Let $A$ be a torsion-free abelian group and $B$ be an injective abelian group. Then we can define new versions of cohomology and homology:
$$
\begin{align\*}
F^n(X,U) &= H^n(X,U) \otimes A \\
G\_n(X,U) &= Hom(H^n(X,U), B)
\end{align\*}
$$
Since $(-) \otimes A$ and $Hom(-,B)$ are functors that preserve exactness, all of the Eilenberg-Steenrod axioms *except* additivity automatically follow. You get a new "non-additive" cohomology or homology theory with $F^0(pt) = A$ and $G\_0(pt) = B$.


For example, if $A$ is infinitely generated free abelian then the wedge axiom fails for $F^\*$ because, for example, the first cohomology of $\bigvee S^1$ is $(\prod \Bbb Z) \otimes A$ instead of $\prod A$.


A more important example is typically furnished by $p$-adically completed homology (and similar theories, such as Morava $E$-theories). There, the homology theory should be viewed as taking values in a category of "complete" groups rather than the category of groups. A discussion of this appears in Appendix A of [this paper of Barthel-Frankland](https://arxiv.org/pdf/1311.7123.pdf).


