  We quantify the pointwise doubling properties of self-similar measures using
the notion of pointwise Assouad dimension. We show that all self-similar
measures satisfying the open set condition are pointwise doubling in a set of
full Hausdorff dimension, despite the fact that they can in general be
non-doubling in a set of full Hausdorff measure. More generally, we carry out
multifractal analysis by determining the Hausdorff dimension of the level sets
of the pointwise Assouad dimension.
