  Exceptional points (EPs) are truly non-Hermitian (NH) degeneracies where
matrices become defective. The order of such an EP is given by the number of
coalescing eigenvectors. On the one hand, most work focusses on studying
$N$th-order EPs in $N\leq4$-dimensional NH Bloch Hamiltonians. On the other
hand, some works have remarked on the existence of EPs of orders scaling with
systems size in models exhibiting the NH skin effect. In this letter, we
introduce a new type of EP and provide a recipe on how to realize EPs of
arbitrary order not scaling with system size. We introduce a generalized
version of the paradigmatic Hatano-Nelson model with longer-range hoppings. The
EPs existing in this system show remarkable physical features: Their associated
eigenstates are localized on a subset of sites and are exhibiting the NH skin
effect. Furthermore, the EPs are robust against generic perturbations in the
hopping strengths as well as against a specific form of on-site disorder.
