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.