Using the state-of-the-art SKA precursor, the MeerKAT radio telescope, we explore the limits to precision pulsar timing of millisecond pulsars achievable due to pulse stochasticity (jitter). We report new jitter measurements in 15 of the 29 pulsars in our sample and find that the levels of jitter can vary dramatically between them. For some, like the 2.2~ms pulsar PSR J2241--5236, we measure an implied jitter of just $sim$ 4~ns/hr, while others like the 3.9~ms PSR J0636--3044 are limited to $sim$ 100 ns/hr. While it is well known that jitter plays a central role to limiting the precision measurements of arrival times for high signal-to-noise ratio observations, its role in the measurement of dispersion measure (DM) has not been reported, particularly in broad-band observations. Using the exceptional sensitivity of MeerKAT, we explored this on the bright millisecond pulsar PSR J0437--4715 by exploring the DM of literally every pulse. We found that the derived single pulse DMs vary by typically 0.0085 cm$^{-3}$ pc from the mean, and that the best DM estimate is limited by the differential pulse jitter across the band. We postulate that all millisecond pulsars will have their own limit on DM precision which can only be overcome with longer integrations. Using high-time resolution filterbank data of 9 $mu$s, we also present a statistical analysis of single pulse phenomenology. Finally, we discuss optimization strategies for the MeerKAT pulsar timing program and its role in the context of the International Pulsar Timing Array (IPTA).
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