  We introduce the hydra continued fractions, as a generalization of the
Rogers-Ramanujan continued fractions, and give a combinatorial interpretation
in terms of shift-plethystic trees. We then show it is possible to express them
as a quotient of m-distinct partition generating functions, and in its dual
form as a quotient of the generating functions of compositions with contiguous
rises upper bounded by m-1. We obtain new generating functions for compositions
according to their local minima, for partitions with a prescribed set of rises,
and for compositions with prescribed sets of contiguous differences.
