  The dynamical discrete web (DyDW),introduced in recent work of Howitt and
Warren, is a system of coalescing simple symmetric one-dimensional random walks
which evolve in an extra continuous dynamical time parameter \tau. The
evolution is by independent updating of the underlying Bernoulli variables
indexed by discrete space-time that define the discrete web at any fixed \tau.
In this paper, we study the existence of exceptional (random) values of \tau
where the paths of the web do not behave like usual random walks and the
Hausdorff dimension of the set of exceptional such \tau. Our results are
motivated by those about exceptional times for dynamical percolation in high
dimension by H\"{a}ggstrom, Peres and Steif, and in dimension two by Schramm
and Steif. The exceptional behavior of the walks in the DyDW is rather
different from the situation for the dynamical random walks of Benjamini,
H\"{a}ggstrom, Peres and Steif. For example, we prove that the walk from the
origin S^\tau_0 violates the law of the iterated logarithm (LIL) on a set of
\tau of Hausdorff dimension one. We also discuss how these and other results
extend to the dynamical Brownian web, the natural scaling limit of the DyDW.
