  A certain two-dimensional lattice model with nearest and next-nearest
neighbor interactions is known to have a limit-periodic ground state. We show
that during a slow quench from the high temperature, disordered phase, the
ground state emerges through an infinite sequence of phase transitions. We
define appropriate order parameters and show that the transitions are related
by renormalizations of the temperature scale. As the temperature is decreased,
sublattices with increasingly large lattice constants become ordered. A rapid
quench results in glass-like state due to kinetic barriers created by
simultaneous freezing on sublattices with different lattice constants.
