TITLE: (Co)homology theories not satisfying the wedge axiom
QUESTION [8 upvotes]: 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. 

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

REPLY [7 votes]: 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.