  Electronic structure methods built around double-electron excitations have a
rich history in quantum chemistry. However, it seems to be the case that such
methods are only suitable in particular situations and are not naturally
equipped to simultaneously handle the variety of electron correlations that
might be present in chemical systems. To this end, the current work seeks a
computationally efficient, low-rank, "ultimate" coupled cluster method based
exclusively on $T_2$ and its products which can effectively emulate more
"complete" methods that explicitly consider higher-rank, $T_{2m}$ operators. We
introduce a hierarchy of methods designed to systematically account for higher,
even order cluster operators - like $T_4, T_6, \cdots, T_{2m}$ - by invoking
tenets of the factorization theorem of perturbation theory and
expectation-value coupled cluster theory. It is shown that each member within
this methodological hierarchy is defined such that both the wavefunction and
energy are correct through some order in many-body perturbation theory (MBPT),
and can be extended up to arbitrarily high orders in $T_2$. The efficacy of
such approximations are determined by studying the potential energy surface of
several prototypical systems that are chosen to represent both non-dynamic,
static, and dynamic correlation regimes. We find that the proposed hierarchy of
augmented $T_2$ methods essentially reduce to standard CCD for problems where
dynamic electron correlations dominate, but offer improvements in situations
where non-dynamic and static correlations become relevant. A notable highlight
of this work is that the cheapest methods in this hierarchy - which are correct
through fifth-order in MBPT - consistently emulate the behavior of the
$\mathcal{O}(N^{10})$ CCDQ method, yet only require a $\mathcal{O}(N^{6})$
algorithm by virtue of factorized intermediates.
