  To any graph we associate a sequence of integers called the gonality sequence
of the graph, consisting of the minimum degrees of divisors of increasing rank
on the graph. This is a tropical analogue of the gonality sequence of an
algebraic curve. We study gonality sequences for graphs of low genus, proving
that for genus up to $5$, the gonality sequence is determined by the genus and
the first gonality. We then prove that any reasonable pair of first two
gonalities is achieved by some graph. We also develop a modified version of
Dhar's burning algorithm more suited for studying higher gonalities.
