jats:titleAbstract</jats:title>jats:pWe give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao’s method for constructing elements in the finite field jats:inline-formulajats:alternativesjats:tex-math$${\mathbb {F}}_{q^n}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:msubmml:miF</mml:mi>mml:msupmml:miq</mml:mi>mml:min</mml:mi></mml:msup></mml:msub></mml:math></jats:alternatives></jats:inline-formula> whose orders are larger than any polynomial in jats:italicn</jats:italic> when jats:italicn</jats:italic> becomes large. Additionally, we discuss the finiteness of polynomials which translate a given finite set of polynomials to become multiplicatively dependent.</jats:p>