  The quantum Rabi model describing the fundamental interaction between light
and matter is a cornerstone of quantum physics. It predicts exotic phenomena
like quantum phase transitions and ground-state entanglement in the
ultrastrong-coupling (USC) regime, where coupling strengths are comparable to
subsystem energies. Despite progress in many experimental platforms, the few
experiments reaching USC have been limited to spectroscopy: demonstrating USC
dynamics remains an outstanding challenge. Here, we employ a circuit QED chip
with moderate coupling between a resonator and transmon qubit to realise
accurate digital quantum simulation of USC dynamics. We advance the state of
the art in solid-state digital quantum simulation by using up to 90
second-order Trotter steps and probing both subsystems in a combined Hilbert
space dimension $\sim80$, demonstrating the Schr\"odinger-cat like entanglement
and build-up of large photon numbers characteristic of deep USC. This work
opens the door to exploring extreme USC regimes, quantum phase transitions and
many-body effects in the Dicke model.
