In algebraic dynamics, we wish to understand the integer sequence given by the degrees of iterates of a dominant rational self-map. Typical examples include the Fibonacci sequence and other linear recurrence sequences, though not all such degree sequences satisfy a finite recurrence. Their growth is captured by the dynamical degree, an invariant governing the topological and arithmetic complexity of the dynamical system. I will present algebraic and analytic results concerning the generating series of these sequences in the toric setting, including an affirmative answer to Jason Bell’s question on the transcendence of the reduction of these series modulo primes.