Let
g(n)=u(n,n)+1. Then
g is a total
function, but it cannot itself be
primitive recursive, because it differs from the
nth of the (countable)
primitive recursive functions on input
n. (Essentially a diagonalization argument).
Note,
g is just a composition of
u with a successor
function. Composition and succession are allowed in
primitive recursive functions, which means that the problematic part of
g preventing it from being
primitive recursive must be
u.