This is a talk in dynamical systems, and especially the dynamics of maps of the unit interval f : [0,1] -> [0,1]. Classical interval dynamics theory says that a transitive, piecewise-monotone map is conjugate to a map with constant absolute value of slope, where the logarithm of the slope is the topological entropy of the system. We show that the story is vastly different for countably piecewise-monotone maps (think of a function with infinitely many local extreme points). For example, we produce such a map with no conjugate map of constant slope on the interval [0,1], but with conjugate maps of constant slope on the "interval" [0,\infty] for every value of slope whose logarithm is greater than or equal to the topological entropy.