Class of functions computable by Coq
Benjamin Werner has proved the mutual interpretability of ZFC with countably many inaccessibles and the Calculus of Inductive Constructions, in his paper Sets in Types, Types in Sets.  

This means, roughly, that any function which can be shown to be total in ZFC with countably many inaccessibles can be defined in Coq. So unless you are a set theorist working on large cardinals, it is unlikely that any computable function you have ever wanted cannot be defined in Coq.