Edit: this doesn't work.
Original: I think that a compact curve in $M_g$ provides an example. In such a family, the length of the shortest geodesic is bounded. Whereas, in a family produced by ramifying over four points in $P^1$, as two ramification points collide, there is a loop whose image downstairs stays near that pair of points, looping about them several times, and thus has arbitrarily small length. There's probably a variant of this argument with Mumford-Tate groups.
Added:
Jason objects that branch points downstairs can collide without ramification points upstairs colliding. This is true, but easy to patch. There are four branch points and some pairs of them might collide without the ramification points colliding, but if all pairs collide without the ramification points colliding, then they must permute different sets of sheets and the curve is not (geometrically) connected.
Jordan objects that even if the ramification points collide, the curve may remain smooth. In particular, if the curve is a $d$-fold cover of $P^1$ and all the monodromy is a power of a fixed $d$-cycle, if two points labeled by $a$ and $b$ so that all three of $a$, $b$, and $a+b$ are relatively prime to $d$, then the family is smooth. If this is true for all pairs of collisions, this gives a complete family of curves that map to $P^1$ with only four branch points, contradicting my claim. In particular, $d=5$ and $1,1,1,2$ is a complete family of genus $4$ curves.
Such examples don't exist for smaller $d$. Thus a complete curve in $M_3$ (which exist, right?) is a good candidate for not having a map to $P^1$ with $4$ branch points. Collisions of branch points with other forms of monodromy are harder for me to understand, but they look more likely to result in degeneration.