  In this article, we show that for any non-isotrivial family of abelian
varieties over a rational base with big monodromy, those members that have
adelic Galois representation with image as large as possible form a density-$1$
subset. Our results can be applied to a number of interesting families of
abelian varieties, such as rational families dominating the moduli of Jacobians
of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we
prove that for any dimension $g \geq 3$, there are infinitely many abelian
varieties over $\mathbb Q$ with adelic Galois representation having image equal
to all of $\operatorname{GSp}_{2g}(\widehat{\mathbb Z})$.
