  We consider several different models for generating random fractals including
random self-similar sets, random self-affine carpets, and fractal percolation.
In each setting we compute either the \emph{almost sure} or the \emph{Baire
typical} Assouad dimension and consider some illustrative examples. Our results
reveal a common phenomenon in all of our models: the Assouad dimension of a
randomly generated fractal is generically as big as possible and does not
depend on the measure theoretic or topological structure of the sample space.
This is in stark contrast to the other commonly studied notions of dimension
like the Hausdorff or packing dimension.
