  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).
