Status of the classification of non-finitely generated abelian groups.

From the Wikipedia on abelian groups:

By contrast, classification of general infinitely-generated abelian groups is far from complete.

How far are we from a classification exactly? It seems like divisible groups have been classified. Which cases are left which we haven't? What is the nature of these unknown cases that makes them so hard to understand? Do we have examples of really "out there" infinitely-generated groups that don't fit into any of the current categories?

One problem with classification of objects of unrestricted size/complexity is set theoretic, and will be related to such strong set theoretic axioms related to accessible cardinals, measurable cardinals, and Vopenka's principle, and may depend on the continuum hypothesis and (certainly) by the axiom of choice. While I'm not an expert on these issues, I'll try to give a quick view of the difficulties.

For abelian groups, you can construct quite wild beasts using Zorn's Lemma. The general strategy will be to first construct some huge abelian group (say the product or coproduct of abelian groups indexed by a set of huge cardinality) and then apply Zorn's Lemma to obtain a maximal subgroup. Such subgroups typically can't be described directly and so would be very hard to study or classify.

Other ways of constructing very big objects is by taking ultraproducts of abelian groups. Here the construction relies on a choice of a principal ultrafilter, a highly non-constructive object. These objects are somewhat more tractable but still the immense freedom in choosing the cardinality of the indexing set, the ultrafilter, and the constituent groups makes this extremely hard.

Somewhat more systematically, since any abelian group is a quotient of a free abelian group, the complete classification requires the classification of the subgroups of free abelian groups. If no size restriction is placed on the number of generators of the free abelian groups again set theoretic subtleties immediately manifest themselves. For instance, taking a free abelian group on a set of huge cardinality, classifying the maximal subgroups in it will be very hard.

The book

L.Fuchs. Infinite Abelian Groups, Vol. I,II. Academic Press, 1970,1973

has a lot of unresolved problems. By them you can get an idea about actual directions in the theory of Abelian groups.