  Dobi\'nski set $\mathcal{D}$ is an exceptional set for a certain infinite
product identity, whose points are characterized as having exceedingly good
approximations by dyadic rationals. We study the Hausdorff dimension and
logarithmic measure of $\mathcal{D}$ by means of the Mass Transference
Principle and by the construction of certain appropriate Cantor-like sets,
termed willow sets, contained in $\mathcal{D}$.
