In (one version of) the story of Hercules and the Lernaean Hydra, every time Hercules chopped off one of the hydra's heads, the hydra regenerated two heads in its place. But, let's consider instead a hydra with more mathematically complex biology. The hydra's heads form a tree, and for the $n$th head that Hercules chops, the hydra produces $n$ copies of whatever else was growing from the base of the neck, at the same point. This hydra seems pretty undefeatable; at every stage it grows more and more heads. However, I will show that not only is the hydra defeatable, any strategy that Hercules chooses to defeat it will eventually work. In doing so, I'll introduce the set-theoretic notion of the ordinals, and prove the termination of Goodstein sequences.