We know that M_g is general type for g large enough. In particular, the generic genus-g curve is not contained in a (non-isotrivial) rational family parametrized by P^1. In fact, the high-genus curves I know how to build over C(t) all have low gonality; it's easy to make a hyperelliptic curve y^2 = f(t,x), and with a little more work you can make curves over k(t) which are 3-gonal, 4-gonal, or 5-gonal. But do we know whether there's a curve over C(t) whose gonality is close to the generic value, which is on order g/2?
A colleague suggested that the linear system of the generator of Pic on a K3 with Picard number 1 would be a good place to look for these. I don't immediately see the proof that these guys have big gonality, but it certainly seems reasonable.
So more generally, one might ask -- are there genus-g curves over C(t) which are "Brill-Noether generic," i.e. which are not distinguished from the generic genus-g curve by the presence of any g^r_d?
Note that I am not asking "what do we expect the union of all rational curves on M_g to look like, assuming we believe Lang's conjecture?" -- that question is too intimidating. Rather, I'm asking whether there is some Brill-Noether locus on M_g which MIGHT contain all the rational curves. Having phrased it this way, I imagine this must be a question with some literature attached, but I wasn't able to find it.