  In 2016, Bourgain, Gamburd, and Sarnak proved that Strong Approximation holds
for the Markoff surface in most cases. That is, the modulo $p$ solutions to the
equation $X_1^2+X_2^2+X_3^2=3X_1X_2X_3$ are covered by the integer solutions
for most primes $p$. In this paper, we provide upper bounds on lifts of mod $p$
points of the Markoff surface by analyzing the growth along paths in the
Markoff mod $p$ graphs. Our first upper bound follows the algorithm given in
the paper of Bourgain, Gamburd, and Sarnak, which constructs a path of possibly
long length but where points grow relatively slowly. Our second bound considers
paths in these graphs of short length but possibly large growth. We then
provide numerical evidence and heuristic arguments for how these bounds might
be improved.
