Structured chain of functions

Question

It is a trivial assertion that the binary operation $g(x,y)\mapsto xy$ is associative and also distributive over the binary operation $f(x,y)\mapsto x+y$. We say that $f\leadsto g$ if this occurs, where all the binary operations in consideration are taken over the reals. The question is to determine whether the longest possible chain of functions $$f_1\leadsto f_2\leadsto f_3\leadsto f_4...$$ is finite.