  We investigate plane curves intersecting in at most two unibranched points to
study the algebraic exceptional set appearing in standard conjectures of
diophantine and hyperbolic geometry. Our first result compares the local
geometry of two hypertangent curves, i.e. curves having maximal contact at one
unibranched point. This is applied to fully describe the exceptional set and,
more generally, the hyper-bitangency set, of a plane curve with three
components.
