  The group $\text{Diff}(\mathcal{M})$ of diffeomorphisms of a closed manifold
$\mathcal{M}$ is naturally equipped with various right-invariant Sobolev norms
$W^{s,p}$. Recent work showed that for sufficiently weak norms, the geodesic
distance collapses completely (namely, when $sp\le \text{dim}\mathcal{M}$ and
$s<1$). But when there is no collapse, what kind of metric space is obtained?
In particular, does it have a finite or infinite diameter? This is the question
we study in this paper. We show that the diameter is infinite for strong enough
norms, when $(s-1)p\ge \text{dim}\mathcal{M}$, and that for spheres the
diameter is finite when $(s-1)p<1$. In particular, this gives a full
characterization of the diameter of $\text{Diff}(S^1)$. In addition, we show
that for $\text{Diff}_c(\mathbb{R}^n)$, if the diameter is not zero, it is
infinite.
