  In this paper, we explore the algebraic and geometric structures that arise
from a procedure we dub "gauging the gauge", which involves the promotion of a
certain global, coordinate independent symmetry to a local one. By gauging the
global 1-form shift symmetry in a gauge theory, we demonstrate that the
structure of a Lie algebra crossed-module and its associated 2-gauge theory
arises. Moreover, performing this procedure once again on a 2-gauge theory
generates a 3-gauge theory, based on Lie algebra 2-crossed-modules. As such,
the physical procedure of "gauging the gauge" can be understood mathematically
as a \textit{categorification}. Applications of such higher-gauge structures
are considered, including gravity, higher-energy physics and condensed matter
theory. Of particular interest is the mechanism of \textit{anomaly resolution},
in which one introduces a higher-gauge structure to absorb curvature
excitations. This mechanism has been shown to allow one to consistently gauge
an anomalous background symmetry in QFT.
