  Hausdorff dimension of level sets of generic continuous functions defined on
fractals can give information about the "thickness/narrow cross-sections"
"network" corresponding to a fractal set, $F$. This lead to the definition of
the topological Hausdorff dimension of fractals. Finer information might be
obtained by considering the Hausdorff dimension of level sets of generic
$1$-H\"older-$\alpha$ functions, which has a stronger dependence on the
geometry of the fractal, as displayed in our previous papers. In this paper, we
extend our investigations to the lower and upper box-counting dimension as
well: while the former yields results highly resembling the ones about
Hausdorff dimension of level sets, the latter exhibits a different behaviour.
Instead of "finding narrow-cross sections", results related to upper
box-counting dimension try to "measure" how much level sets can spread out on
the fractal, how widely the generic function can "oscillate" on it. Key
differences are illustrated by giving estimates concerning the Sierpi\'nski
triangle.
