Total functional computable real numbers
The set of higher-order primitive recursive reals is essentially the class of functions $\mathbb{N}\rightarrow\mathbb{N}$ which can be represented by a term $\mathrm{Nat}\rightarrow\mathrm{Nat}$ in Gödel's system T.

Since every such function is total, and every well-typed term in the system can be enumerated effectively, there is a relatively easy proof by diagonalization that there is some computable real which cannot be represented.