There are many examples of gauge theories with disconnected gauge groups, but as far as I know, the non-trivial topological sectors of these theories are beyond the current experimental capabilities.
In order for large gauge transformations to act nontrivially on the states, it must be a symmetry of the Hamiltonian. For example, the Maxwell Hamiltonian depends only on the electric and magnetic fields. In flat space, the electromagnetic $U(1)$ gauge group is connected, thus no large gauge transformations exists. However, if appropriate boundary conditions are provided, for example periodic boundary conditions in one direction, then large gauge transformations exist and are symmetries of the Hamiltonian.
In contrast, the Hamiltonian a charged particle moving under the influence of a background electromagnetic field is only quasi-invariant, only when the gauge transformation is performed also on the wave functions, the spectrum is conserved. In this case, large gauge transformations must be considered as gauge redundancies rather than symmetry. (However, a gas of such particles has an invariant second quantized Hamiltonian, but I don't know how to exploit this fact).
This is the reason why the Aharonov-Bohm system of a particle moving on a circle around a magnetic flux, does not possess large gauge symmetries in spite of the fact that the electromagnetic gauge group is disconnected because $\pi_1(U(1)) = \mathbb{Z}$.
What I am going to describe to you here is a (quite ingenious) experiment suggested by S.-R. Eric Yang. He proposed a modification of the Aharonov-Bohm setting to introduce degenerate energy eigenfunctions related by a large gauge transformation. This experiment seems feasible, but from a google scholar search, there does not seem that this experiment has yet been actually performed.
The trick is to use a spin half particle and add an electric field in the radial direction in order to generate a spin orbit interaction (proportional to: $\vec{\sigma} \cdot \left (\vec{p} – i e \vec{A} \right )$).
The spin orbit term breaks the time reversal invariance, but Yang noticed is that when the Aharonov-Bohm potential is equal to a half flux quantum $A_{\phi} = \frac{1}{2}$, then both kinetic and the gauge orbit interaction terms become invariant under the large gauge transformation $e^{i\phi}$ (which shifts the gauge potential by one quantum) followed by a time reversal transformation. Thus, this transformation is a symmetry of the Hamiltonian. As a consequence there are two degenerate states of opposite spin which are related by the above transformation. These two states should be able to be distinguished between by means of their Berry phases.
Update
I am using the term gauge group for the group of all gauge transformations (including small and large gauge transformations). In our case it is not the one dimensional group $U(1)$ of global gauge transformations, but the infinite group $\mathrm{Map}(S^1, U(1))$ of local gauge transformations. This group is disconnected. Its disconnected part modulo the connected component (which form the group of large gauge transformations) is $\pi_1(S^1)= \mathbb{Z}$ realized by means of transformations of the type $e^{i n \phi}$ for integer $n$. This is essentially the same gauge group addressed by Landsman and Wren in my answer of the attached question in the main text (in their case it is $\mathrm{Map}(S^1, U(n))$ , but since for $G$ semisimple and centerless $\pi_1(G)= 0$ , thus only the $U(1)$ part of $U(n)$ contributes to the large gauge transformations.
As I emphasized in my answer to Friedrich's comment in the attached question; the group of disconnected elements from the unit component (modulo connected component) is the basic definition of large gauge transformation. It is true that when you describe spheres as one point compactifications of flat spaces, then the large gauge transformations become those which do not approach the unit at infinity. You can do this exercise for our case by expressing $S^1$ as one point compactification of $\mathbb{R}$.
The Hamiltonian of the Aharonov-Bohm system is not invariant under large gauge transformation because the theory is only quasi-invariant and not fully invariant. This can be easily checked by direct substitution.
A symmetry of the theory is a transformation commuting with the Hamiltonian without acting on the wave functions. I only emphasized that to tell that the A-B system is not invariant under large gauge transformations.
Your last comment correctly summarises the distinction between symmetry and redundancy.
