  We determine almost sure limits of rescaled intrinsic volumes of the
construction steps of fractal percolation in $\mathbb{R}^d$ for any dimension
$d\geq 1$. We observe a factorization of these limit variables which allows, in
particular, to determine their expectations and covariance structure. We also
show convergence of rescaled expectations and variances of the intrinsic
volumes of the construction steps to expectations and variances of the limit
variables and give rates for this convergence in some cases. These results
significantly extend our previous work that addressed only limits of
expectations of intrinsic volumes.
