(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.

The following example of a non-additive homology theory is due to James and Whitehead, in "Homology with zero Coeffients". It appears as an example in Rudyak's book "On Thom Spectra, Orientability, and Cobordism", pg 65.

Define

$\tilde{h}(X)=\frac{\prod_{n=0}^{\infty} \tilde{H}_n(X)}{\sum_{n=0}^{\infty} \tilde{H}_n(X)}$

where $\tilde{H_*}$ is (reduced) ordinary homology. In particular $\tilde{h}(S^n)=0$ for each $n$ but $\tilde{h}(\vee_{n=0}^\infty\; S^n)\neq 0$. James and Whitehead's paper also contains other examples which may be of interest.