Function that is not Space Constructible
Usually such examples are specifically "tailored", using hard problems.
Perhaps the easiest way is to take the halting problem:
$$HALT=\{\langle M\rangle: M\text{ is a TM that halts on }\epsilon\}$$
and consider the encodings of TMs as numbers, in such a way that no two numbers are encoded to the same TM (e.g. by leading 0s). This can be achieved using unary encoding, for example.

Now, consider the function $f:\mathbb{N}\to \mathbb{N}$ defined as follows:
$f(n)=2n$ if $n$ represents a TM in HALT, and $f(n)=2n+1$ otherwise.

This function is $\theta(n)$, but clearly it is not space constructible, since constructing it would enable you to solve the halting problem.

A similar example can be constructed from a problem which is harder than PSPACE (e.g. a problem complete in EXPSPACE).

However, these example are somewhat "ugly". A more elegant example is the busy beaver function, which is non-computable simply because it "grows too fast".