  In first-passage percolation (FPP), we let $(\tau_v)$ be i.i.d. nonnegative
weights on the vertices of a graph and study the weight of the minimal path
between distant vertices. If $F$ is the distribution function of $\tau_v$,
there are different regimes: if $F(0)$ is small, this weight typically grows
like a linear function of the distance, and when $F(0)$ is large, the weight is
typically of order one. In between these is the critical regime in which the
weight can diverge, but does so sublinearly. We study a dynamical version of
critical FPP on the triangular lattice where vertices resample their weights
according to independent rate-one Poisson processes. We prove that if $\sum
F^{-1}(1/2+1/2^k) = \infty$, then a.s. there are exceptional times at which the
weight grows atypically, but if $\sum k^{7/8} F^{-1}(1/2+1/2^k) <\infty$, then
a.s. there are no such times. Furthermore, in the former case, we compute the
Hausdorff and Minkowski dimensions of the exceptional set and show that they
can be but need not be equal. These results show a wider range of dynamical
behavior than one sees in subcritical (usual) FPP.
