We consider the problem of when the first $n$ iterates of a given rational function over an arbitrary field are multiplicatively independent. This leads to a generalisation of a method of Gao (1999) for constructing elements of high order in finite fields.

Additionally, we present a kind of function field analogue to results of Bombieri, Masser and Zannier (1999), and Dubickas and Sha (2016) on multiplicative dependence of translations of algebraic numbers.