Discussions of large gauge transformations typically only happen in the context of the temporal gauge. After fixing $A_0$, we still have a residual gauge freedom which consists of all time-independent gauge transformations.
These time-independent gauge transformations can be classified according to their behavior at spatial infinity. Famously, there are gauge transformations with nonzero winding number which live in topologically distinct classes. This is especially important for the ground state of gauge theories. All gauge transformations of $A_\mu=0$ are ground state configurations too. But since our gauge transformations fall into topologically distinct classes, the (pure gauge) gauge potential configurations fall into topologically distinct classes too. These observations lead, for example, to the standard periodic picture of the QCD vacuum.
However, choosing the temporal gauge is certainly not mandatory. Instead, it should be possible to choose a gauge like the axial gauge which leaves no residual gauge freedom.
But then, what about large gauge transformations? Does any analogue exist in such "physical" gauges?