  We demonstrate the existence of topologically stable unpaired exceptional
points (EPs), and construct simple non-Hermitian (NH) tight-binding models
exemplifying such remarkable nodal phases. While fermion doubling, i.e. the
necessity of compensating the topological charge of a stable nodal point by an
anti-dote, rules out a direct counterpart of our findings in the realm of
Hermitian semimetals, here we derive how noncommuting braids of complex energy
levels may stabilize unpaired EPs. Drawing on this insight, we reveal the
occurrence of a single, unpaired EP, manifested as a non-Abelian monopole in
the Brillouin zone of a minimal three-band model. This third-order degeneracy
represents a sweet spot within a larger topological phase that cannot be fully
gapped by any local perturbation. Instead, it may only split into simpler
(second-order) degeneracies that can only gap out by pairwise annihilation
after having moved around inequivalent large circles of the Brillouin zone. Our
results imply the incompleteness of a topological classification based on
winding numbers, due to non-Abelian representations of the braid group
intertwining three or more complex energy levels, and provide insights into the
topological robustness of non-Hermitian systems and their non-Abelian phase
transitions.
