  We say that $(x,y,z)\in Q^3$ is an associative triple in a quasigroup $Q(*)$
if $(x*y)*z=x*(y*z)$. Let $a(Q)$ denote the number of associative triples in
$Q$. It is easy to show that $a(Q)\ge |Q|$, and we call the quasigroup
maximally nonassociative if $a(Q)= |Q|$. It was conjectured that maximally
nonassociative quasigroups do not exist when $|Q|>1$. Dr\'apal and Lison\v{e}k
recently refuted this conjecture by proving the existence of maximally
nonassociative quasigroups for a certain infinite set of orders $|Q|$. In this
paper we prove the existence of maximally nonassociative quasigroups for a much
larger set of orders $|Q|$. Our main tools are finite fields and the Weil bound
on quadratic character sums. Unlike in the previous work, our results are to a
large extent constructive.
