We study automata-theoretic classes of string-to-string functions whose output may grow faster than linearly in their input. Our central contribution is to introduce a new such class, with polynomial growth and three equivalent definitions: the smallest class containing regular functions and closed under a "composition by substitution" operation; a restricted variant of pebble transducers; a λ-calculus with a linear type system. As their name suggests, these comparison-free polyregular functions form a subclass of polyregular functions; we prove that the inclusion is strict. Other properties of our new function class that we show are incomparability with HDT0L transductions and closure under composition. Finally, we look at the recently introduced layered streaming string transducers (SSTs), or equivalently k-marble transducers. We prove that a function can be obtained by composing such transducers together if and only if it is polyregular, and that k-layered SSTs (or k-marble transducers) are equivalent to a corresponding notion of (k+1)-layered HDT0L systems.