  Semiconductor moir\'e superlattices comprise an array of artificial atoms and
provide a highly tunable platform for exploring novel electronic phases. We
introduce a theoretical framework for studying moir\'e quantum matter that
treats intra-moir\'e-atom interactions exactly and is controlled in the limit
of large moir\'e period. We reveal an abundance of new physics arising from
strong electron interactions when there are multiple electrons within a moir\'e
unit cell. In particular, at filling factor $n=3$, the Coulomb interaction
within each three-electron moir\'e atom leads to a three-lobed ``Wigner
molecule''. When their size is comparable to the moir\'e period, the Wigner
molecules form an emergent Kagome lattice. Our work identifies two universal
length scales characterizing the kinetic and interaction energies in moir\'e
materials and demonstrates a rich phase diagram due to their interplay.
