Examples include gravity potential, electrostatic potential.
Or, a vector field describing a divergence free vector field (field with only closed field lines) that is its curl: magnetic potential.
Because the physically observable field is a spatial derivative of its potential, adding an arbitrary constant field to it  a gauge transformation  will not change anything in the physics of a system. This is called gauge invariance[?].
In quantum theory, we can identify the potential of a field with the wave function of the intermediary particle associated with that field, like the photon for the electromagnetic field, etc.
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