  Meromorphic connections on Riemann surfaces originate and are closely related
to the classical theory of linear ordinary differential equations with
meromorphic coefficients. Limiting behaviour of geodesics of such connections
has been studied by e.g. Abate, Bianchi and Tovena in relation with generalized
Poincar\'{e}-Bendixson theorems. At present, it seems still to be unknown
whether some of the theoretically possible asymptotic behaviours of such
geodesics really exist. In order to fill the gap, we use the branched affine
structure induced by a Fuchsian meromorphic connection to present several
examples with geodesics having infinitely many self-intersections and quite
peculiar omega-limit sets.
