  We investigate the out-of-equilibrium properties of a system of interacting
bosons in a ring lattice. We present a Floquet driving that induces clockwise
(counterclockwise) circulation of the particles among the odd (even) sites of
the ring which can be mapped to a fully connected model of clocks of two
counter-rotating species. The clock-like motion of the particles is at the core
of a period-$n$ discrete time crystal where $L=2n$ is the number of lattice
sites. In presence of a "staircase-like" on-site potential, we report the
emergence of a second characteristic timescale in addition to the period
$n$-tupling. This new timescale depends on the microscopic parameters of the
Hamiltonian and is incommensurate with the Floquet period, underpinning a
dynamical phase we call 'time quasicrystal'. The rich dynamical phase diagram
also features a thermal phase and an oscillatory phase, all of which we
investigate and characterize. Our simple, yet rich model can be realized with
state-of-the-art ultracold atoms experiments.
