  In this article, we show that in each of four standard families of
hyperelliptic curves, there is a density-$1$ subset of members with the
property that their Jacobians have adelic Galois representation with image as
large as possible. This result constitutes an explicit application of a general
theorem on arbitrary rational families of abelian varieties to the case of
families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit
examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose
Jacobians have such maximal adelic Galois representations.
