  Mott insulators, in which strong Coulomb interactions fully localize
electrons on single atomic sites, play host to an incredibly rich and exciting
array of strongly correlated physics. One can naturally extend this concept to
cluster Mott insulators, wherein electrons localize not on single atoms but
across clusters of atoms, forming ``molecules in solids''. The resulting
localized degrees of freedom incorporate the full spectrum of electronic
degrees of freedom, spin, orbital, and charge. These serve as the building
blocks for cluster Mott insulators, and understanding them is an important
first step toward understanding the many-body physics that emerges in candidate
cluster Mott insulators. Here, we focus on elementary building blocks,
neglecting some of the complexity present in real materials which can often
obfuscate the underlying principles at play. Through an extensive set of exact
theoretical calculations on clusters of varying geometry, number of orbitals,
and number of electrons, we uncover some of the basic organizing principles of
cluster Mott phases, particularly when interactions dominate and negate a
simple single-particle picture. These include, for example, the identification
of an additional ``cluster Hund's rule'', of cluster ground states that are
best understood from a purely interacting perspective, and of several localized
degrees of freedom which are protected by an unusual combination of discrete
spatial or orbital symmetries. Finally, we discuss the impact of adding
additional terms, relevant to material candidates, on the phase diagrams
presented throughout, as well as the potential next steps in the journey to
building a more complete picture of cluster Mott insulators.
