  Given an elliptic curve $E$ over a global function field $K$, the Galois
action on the $n$-torsion points of $E$ gives rise to a mod-n Galois
representation $\rho_{E,n}$. For $K$ satisfying some mild conditions, we show
that the set of $E$ for which $\rho_{E,n}$ is as large as possible for all $n$,
has density $1$.
