Open quantum systems typically relax to a stationary state over time. However, they can also exhibit various non-stationary states, such as continuous time crystals and steady-state chaotic phases, arising from dissipative phase transitions. Periodic non-stationary states are referred to as continuous time crystals because they spontaneously break the continuous time-translation symmetry of the underlying master equation. We demonstrate that such time crystal phases can emerge in a Bose-Hubbard dimer due to the competition between coherent and incoherent hopping processes. The chaotic phase, in contrast, manifests as aperiodic motion in time.We demonstrate that such a chaotic phase arises from the interplay of two competing mechanisms that drive different time crystalline orders in coupled collective spin systems. This competition induces frustration within the system, breaking periodicity and giving rise to chaotic dynamics that persist indefinitely in open quantum systems.