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In gauge theory two configurations $\phi_1, \phi_2$ of gauge fields may be different and still be equivalent: there may be a gauge transformation $\lambda \colon \phi_1 \to \phi_2$ between them.
In higher gauge theory also gauge transformations themseves may be different but still equivalent: if there is a gauge-of-gauge transformation $\rho \colon \lambda_1 \to \lambda_2$ between them.
These higher order gauge transformations are maybe best known in the physics literature in terms of their infinitesimal approximation, the BRST complex: here the gauge transformations correspond to ghost fields and the gauge-of-gauge transformations to ghost-of-ghost fields.
A basic example of a gauge field that has higher order gauge transformations is the B-field. But also magnetic current, if described properly, exhibits higher gauge transformations, see at Dirac charge quantization.
For more see at geometry of physics.