I have not dealt professionally with set theory, so excuse me if my way of formulating this question does not completely follow standard terminology. Actually, my question is about whether or not the following idea can be formalised:
The question is whether it makes sense to talk about a model for ZFC where the number of real numbers is maximal. I know that Paul Cohen has proved that there exist models for ZFC where $\Bbb R$ has an arbitrarily large cardinality. However, as Löwenheim–Skolem shows us, cardinality within a model is not the same as the cardinality of the number of elements in a model. So would it be possible to construct a model $M$ for ZFC in which, for any model $M'$ and a real number $r'$ in $M'$, there exists a real number $r$ in $M$ which has the same numerical value, i.e. the same digits?
The reason I would think this could actually be the case is this: We choose to construct the real numbers $\Bbb R$ as the set of maps $r\colon\Bbb Z\to\{0,1,2,\ldots, 9\}$ with $r(n) = 0$ for all sufficiently large $n$. Intuitively, $r_n:=r(n)$ is the $n$th digit of $r$, i.e. $r = r_N\ldots r_0.r_{-1} r_{-2}\ldots$ for some $N\ge 0$. Of course we shall have to apply an equivalence relation because, for instance, $0.999999\ldots$ must be equal to $1$. Now, this is where it goes vague, but what we do is to include any such sequence of numbers in $\{0,1,2,\ldots,9\}$. I mean literally that any such combination must correpsond to a well-defined map $\Bbb Z\to \{0,1,2,\ldots, 9\}$.
But is this idea of "any combination" possible to formalise in model theory? In this case, is my maximal model for $\Bbb R$ possible to construct formally?
What you seem to be really asking is whether or not there is a model which includes all the real numbers.
Under certain, perhaps even reasonable, assumptions yes. There is such model. Note that this model, however, must be uncountable. And since we want set-models, class models (the universe of sets, for example) are out of the picture. Otherwise, $V$ itself is such model.
As for "construct", since we cannot construct any models of set theory, the term is a bit vague. When we construct a model of $\sf ZFC+\lnot CH$ we begin with a model of $\sf ZFC$, we don't just make one out of thin air. But here you require more, and so the starting conditions require more.
If there exists an inaccessible cardinal $\kappa$, for example, then $V_\kappa$ is a model of $\sf ZFC$ which includes all the real numbers.
Finally, let me point out that in different models of $\sf ZFC$ real numbers can behave differently. For example, if there is any model of $\sf ZFC$ then there is such $M$ such that $\{x\mid M\models x\text{ is an integer}\}$ is an uncountable set, as large as you'd like it to be. If $M$ has more integers than the universe has real numbers, then there are real numbers in $M$ -- or rather objects which $M$ thinks of as real numbers, which cannot be matched by actual real numbers.
So instead the question should be reduced to models which agree with the universe about what are real numbers. Or to make it simpler, to transitive models of $\sf ZFC$. And in this case, there are either such models or there are not. Depending on your starting assumptions.