  We derive exact formulas for circular Wilson loops in the $\mathcal{N}=4$ and
$\mathcal{N}=2^{* }$ theories with gauge groups $U(N)$ and $SU(N)$ in the
$k$-fold symmetrized product representation. The formulas apply in the limit of
large $k$ and small Yang-Mills coupling $g$, with fixed effective coupling
$\kappa\equiv g^2k$, and for any finite $N$. In the $SU(2)$ and $U(2)$ cases,
closed analytic formulas are obtained for any $k$, while the $1/k$ series
expansions are asymptotic. In the $N\gg 1$ limit, with $N\ll k$, there is an
overlapping regime where the formulas can be confronted with results from
holography. Simple formulas for correlation functions between the $k$-symmetric
Wilson loops and chiral primary operators are also given.
