  We revisit the regularity theory of Escauriaza, Seregin, and \v{S}ver\'ak for
solutions to the three-dimensional Navier-Stokes equations which are uniformly
bounded in the critical $L^3_x(\mathbf{R}^3)$ norm. By replacing all
invocations of compactness methods in these arguments with quantitative
substitutes, and similarly replacing unique continuation and backwards
uniqueness estimates by their corresponding Carleman inequalities, we obtain
quantitative bounds for higher regularity norms of these solutions in terms of
the critical $L^3_x$ bound (with a dependence that is triple exponential in
nature). In particular, we show that as one approaches a finite blowup time
$T_*$, the critical $L^3_x$ norm must blow up at a rate $(\log\log\log
\frac{1}{T_*-t})^c$ or faster for an infinite sequence of times approaching
$T_*$ and some absolute constant $c>0$.
