Functions that typed lambda calculus cannot compute
A nice example is given by Godelization: in lambda calculus, the only thing you can do with a function is to apply it. As a result, there is no way to write a closed function of type $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$, which takes a function argument and returns a Godel code for it. 

Adding this as an axiom to Heyting arithmetic is usually called "the constructive Church thesis", and is a strongly anti-classical axiom. Namely, it is consistent to add it to HA, but not to Peano arithmetic! (Basically, it is a classical fact that every Turing machine halts or not, and there is no computable function that can witness this fact.)