  We study the \emph{upper regularity dimension} which describes the extremal
local scaling behaviour of a measure and effectively quantifies the notion of
\emph{doubling}. We conduct a thorough study of the upper regularity dimension,
including its relationship with other concepts such as the Assouad dimension,
the upper local dimension, the $L^q$-spectrum and weak tangent measures. We
also compute the upper regularity dimension explicitly in a number of important
contexts including self-similar measures, self-affine measures, and measures on
sequences.
