Definition of time-constructible function

The basic use of time-constructibility (and space-constructibility) is to clock the time a machines runs (or space it uses), i.e. we want to simulate a machine only for $t(n)$ steps on an input of length $n$, only using $O(t(n))$ steps. To do this, we need to compute the value of $t(n)$ in time $O(t(n))$. If $t(n)$ cannot be computed in time $O(t(n))$ the total running time of simulation will not be $O(n)$.

An example is when we want to prove a hierarchy theorem. We need to simulate all machines in the smaller class in the time bound of the larger class.

There is no self-reference. Fix a function, e.g. $n^2$. This is function, we are not talking about how it is computed. Now the question is given $n$, we want to compute the value of $n^2$ (both input and output in binary). What is the complexity of this problem? Well, it is $O(n^2)$. So $n^2$ is time-constructible. The function $n^2$ is used twice (once as function we want to compute and once as the time bound) but it is not a self-reference.

Update:

Is there a function which is not time-constructible?

Yes, there are. E.g. any non-computable function is not time constructible. But I think you mean are there computable functions which are not time constructible? The answer is still yes, but giving examples is not very easy because most functions that we deal with in practice turn out to be time constructible.