  We study the thick points of branching Brownian motion and branching random
walk with a critical branching mechanism, focusing on the critical dimension $d
= 4$. We determine the exponent governing the probability to hit a small ball
with an exceptionally high number of pioneers, showing that this has a
second-order transition between an exponential phase and a
stretched-exponential phase at an explicit value ($a = 2$) of the thickness
parameter $a$. We apply the outputs of this analysis to prove that the
associated set of thick points $\mathcal{T}(a)$ has dimension $(4-a)_+$, so
that there is a change in behaviour at $a=4$ but not at $a = 2$ in this case.
Along the way, we obtain related results for the nonpositive solutions of a
boundary value problem associated to the semilinear PDE $\Delta v = v^2$ and
develop a strong coupling between tree-indexed random walk and tree-indexed
Brownian motion that allows us to deduce analogues of some of our results in
the discrete case.
  We also obtain in each dimension $d\geq 1$ an infinite-order asymptotic
expansion for the probability that critical branching Brownian motion hits a
distant unit ball, finding that this expansion is convergent when $d\neq 4$ and
divergent when $d=4$. This reveals a novel, dimension-dependent critical
exponent governing the higher-order terms of the expansion, which we compute in
every dimension.
