  We initiate a study of distortion elements in the Polish groups
$\mbox{Diff}_+^k(\mathbb{S}^1)$ ($1\leq k<\infty$), as well as
$\mbox{Diff}_+^{1+AC}(\mathbb{S}^1)$, in terms of maximal metrics on these
groups. We classify distortion in the $k=1$ case: a $C^1$ circle diffeomorphism
is $C^1$-undistorted if and only if it has a hyperbolic periodic point. On the
other hand, answering a question of Navas, we exhibit analytic circle
diffeomorphisms with only non-hyperbolic fixed points which are
$C^{1+AC}$-undistorted, and hence $C^k$-undistorted for all $k\geq 2$. In the
appendix, we exhibit a maximal metric on $\mbox{Diff}_+^{1+AC}(\mathbb{S}^1)$,
and observe that this group is quasi-isometric to a hyperplane of $L^1(I)$.
