  We use a single trapped-ion qutrit to demonstrate the violation of an
input-state-independent non-contextuality inequality using a sequence of
randomly chosen quantum non-demolition projective measurements. We concatenate
54 million sequential measurements of 13 observables, and violate an optimal
non-contextual bound by 214 standard deviations. We use the same dataset to
characterize imperfections including signaling and repeatability of the
measurements. The experimental sequence was generated in real time with a
quantum random number generator integrated into our control system to select
the subsequent observable with a latency below 50 {\mu}s, which can be used to
constrain hidden-variable models that might describe our results. The
state-recycling experimental procedure is resilient to noise, self-correcting
and independent of the qutrit state, substantiating the fact that quantumness
is connected to measurements as opposed to designated states.
