Different nuclear burning regimes are predicted on accreting neutron stars (NSs) depending mainly -but not only- on the amount of mass accreted on the NS per unit time and area. We present X-ray observations of a rapidly accreting 11 Hz pulsar in the globular cluster Terzan 5 that reveal four distinct thermonuclear bursting regimes. In one of them we find a steep (inverse cubic) relation between burst recurrence time and inferred mass accretion rate. We also report the discovery and detailed analysis of mHz quasi-periodic oscillations (QPOs) from the same NS, usually interpreted as marginally stable burning. We compare the unique burst and mHz QPO properties of this system to previous observations and to models of burst ignition near the helium burning stability boundary, and find new constraints on the heat sources present in the NS envelope. Finally, we discuss the potential impact of the NS magnetic field and spin on the expected burning regimes in the context of this particular system, which bridges the gap between accreting NSs in high- and low-mass X-ray binaries.