For a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, the fundamental group of $\mathfrak{h}_\text{reg}/W$ (where $\mathfrak{h}$ is the Cartan subalgebra, $\mathfrak{h}_\text{reg}$ is the subset of its regular elements, and $W$ the Weyl group) is the Artin braid group of type $\mathfrak{g}$.

Similarly, if we take a simple and simply-connected Lie group $G$ over $\mathbb{C}$ and consider the orbifold fundamental group of $T_\text{reg}/W$ (where $T$ is the Cartan torus of $G$), it is the affine Artin braid group of the corresponding type.

The references to the proofs to these statements can be found e.g. in Charney-Davis.

Now, take the moduli space of $G$-bundles on an elliptic curve $E$, which is known to be given by $(E\otimes_{\mathbb{Z}}\Lambda) /W$ (cf. Laszlo ) where $\Lambda$ is the coweight lattice of $T$. I believe the fundamental group of this space (minus 'non-regular' elements) is more or less given by the double affine braid group of the corresponding type, but I don't find any explicit reference. Could you give me one?