  This paper is concerned mainly with the macroscopic fractal behavior of
various random sets that arise in modern and classical probability theory.
Among other things, it is shown here that the macroscopic behavior of Boolean
coverage processes is analogous to the microscopic structure of the Mandelbrot
fractal percolation. Other, more technically challenging, results of this paper
include: (i) The computation of the macroscopic dimension of the graph of a
large family of L\'evy processes; and (ii) The determination of the macroscopic
monofractality of the extreme values of symmetric stable processes.
  As a consequence of (i), it will be shown that the macroscopic fractal
dimension of the graph of Brownian motion differs from its microscopic fractal
dimension. Thus, there can be no scaling argument that allows one to deduce the
macroscopic geometry from the microscopic. Item (ii) extends the recent work of
Khoshnevisan, Kim, and Xiao \cite{KKX} on the extreme values of Brownian
motion, using a different method.
