If a configuration space is topologically non-trivial, one can distinguish between "small" gauge transformations which can be smoothly deformed to the identity, and "large" gauge transformations which cannot be smoothly deformed
to the identity because they "wind" around the "handles" of the configuration space.
An important example of topologically non-trivial configuration space is a non-Abelian gauge theory. In the Abelian case the transverse gauge condition $\partial \cdot A=0$ is sufficient to remove the degeneracy. While in non-Abelian case, as Gribov showed, there are distinct transverse conﬁgurations $A\neq A^\prime$ such that $\partial \cdot A=\partial \cdot A^\prime=0$ and these conﬁgurations are connected with each other by "large" gauge transformations. This topological non-triviality of the configuration space has an important impact on QCD dynamics: http://arxiv.org/abs/1202.1491 (The Gribov problem and QCD dynamics, by N. Vandersickel and D. Zwanziger).
Chern-Simons theory provides another example where "large" gauge transformations have an important role: http://arxiv.org/abs/hep-th/9902115 (Aspects of Chern-Simons Theory, by G. V. Dunne).