Taffy is a type of candy made by repeated 'pulling' (stretching and folding) a mass of heated sugar. The purpose of pulling is to get air bubbles into the taffy, which gives it a nicer texture. Until the late 19th century, taffy was pulled by hand, an arduous task. The early 20th century saw an avalanche of new devices to mechanise the process. These devices have fascinating connections to the topological dynamics of surfaces, in particular with pseudo-Anosov maps. "Special" algebraic integers such as the Golden ratio and the lesser-known Silver ratio make an appearance, as well as more exotic numbers. Finally, we ask the question: can modern mathematics help us improve on these designs?