TITLE: A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom?
QUESTION [21 upvotes]: Let us agree on the following: a "homology theory" means a functor $h_*$ from the category of pointed CW complexes to the category of graded abelian groups, together with natural isomorphisms $h_{*+1}(\Sigma X)\cong h_*(X)$, such that the functor $h_*$ is homotopy invariant, sends a cofiber sequence to an exact sequence, and satisfies Milnor's additivity axiom: the natural map to the homology of a wedge from the direct sum of the homologies is an isomorphism.
Adding Milnor's axiom to the list is a reasonable thing to do, partly because it guarantees uniqueness in the following sence: a morphism of homology theories (satisfying Milnor's axiom) which induces an isomorphism on the coefficients, is itself an isomorphism of homology theories.
Now enter the world of spectra. A spectrum $E$ defines a homology theory via $E_*(X)=\pi_*(E\wedge X)$. Such a homology theory satisfies an axiom a priori stronger than Milnor's additivity: it satisfies the

Direct limit axiom: The natural map to the homology of $X$ from the colimit of the homologies of the finite subcomplexes of $X$ is an isomorphism.

Reciprocally, a homology theory satisfying this axiom comes from a spectrum in the way defined above (see Adams' Stable Homotopy and Generalised Homology, pp. 199-200 and Adams "A Variant of E.H. Brown's representability theorem".)
So the homology theories given by spectra are a bit better behaved than a homology theory as is usually defined. The question is:

Could you give an example of a homology theory which does not satisfy the direct limit axiom?

I first asked this question on a comment here, and Mark Grant pointed out that by Hatcher's exercise 4.F.1, Milnor's additivity implies the direct limit axiom for countable complexes, so we'd need to go to uncountable ones.

REPLY [2 votes]: Edit: As explained in the comments, this is not an answer to the question. I 
leave it as it might at least serve as an intuition for why the direct sum axiom plus excision should be enough to imply the direct limit axiom.
Working $\infty$-categorically, we may think of a homology theory as a functor from spaces to spectra. The axioms about the suspension isomorphism and cofiber sequences are equivalent to preservation of pushouts. Now, a functor which preserves pushouts and all (small) coproducts, preserves all (small) colimits. This is proposition 4.4.2.7 in HTT by Lurie. 
Remark: Actually, we can reverse the logic by saying that we just want our the functor to preserve all (small) colimits. One can break this into (1) preservation of finite colimits and (2) preservation of filtered colimits. One can also replace (2) by (2') preservation of coproducts. Additionally, one can replace (1) by (1') preservation of pushouts. Finally, one can test that a map of spectra is an equivalence by checking the induced map on the homotopy groups and for a pushout these can be described by a long exact sequence. This allows us to reformulate everything in classical terms (i.e. the Eilenberg-Steenrod axioms).