  We investigate the analogues, in $\mathbb{F}_q[t]$, of highly composite
numbers and the maximum order of the divisor function, as studied by Ramanujan.
In particular, we determine a family of highly composite polynomials which is
not too sparse, and we use it to compute the logarithm of the maximum of the
divisor function at every degree up to an error of a constant, which is
significantly smaller than in the case of the integers, even assuming the
Riemann Hypothesis.
