Example of an infinite dimensional Hilbert space that is not an RKHS

There are no explicit examples of this kind. Indeed, if one could show you such a space $\mathcal H$, then point evaluation $f\mapsto f(x_0)$ would give an explicit example of a discontinuous linear functional on a Hilbert space. And there can be no explicit examples of this kind, because it's consistent with ZF that they do not exist. Only with the Axiom of Choice can one give a (non-constructive) proof of their existence. This is discussed in many places, such as

• Discontinuous linear functional
• Unbounded linear operator defined on $l^2$

So, any time someone gives you a concrete Hilbert space of functions, you can be sure that it's either RKHS, or is not really a space of functions.