a set theoretical completion? Is there such a thing as starting with a weak kind of set theory and making an argument to show that it should be expanded into a stronger set theory such as ZFC? Something similar to going from the rationals to the reals? With number systems, we have evidence which forces us into a stronger system (sqrt of 2). Is there such evidence with set theory?
I mentioned the real numbers because in that situation there is obviously something missing. Is there obviously something missing from "weak" set theories? My interest is: to what extent can we justify the axioms of infinity and choice? Is there a higher level principle (of the "there is something missing" sort) which implies infinity and choice should be part of our system?
 A: While your analogy with the case of rationals $\rightarrow$ reals is (in my opinion) somewhat strained, I think this is a very good question. Perhaps it would be better to recast it as:

What are criteria which might make us want to add axioms to our set theory?

There are three broad cases I can think of:


*

*Accomodating informal set-theoretic arguments. Basically, the idea is that any "reasonable" argument in naive set theory should have a counterpart in the formal set theory. This was a driving force behind adding the axiom of replacement (due to Fraenkel); Zermelo set theory without replacement (ZC) is vastly weaker than ZFC. But this process appears to have ended with ZFC.

*Solving mathematical problems. We might argue, for instance, that the Continuum Hypothesis should be resolvable in set theory. This would mean we need to add axioms, since our current set of axioms are insufficient. Similarly, a desire to develop a robust theory of projective sets leads to large cardinals. This is a major driving force in modern set-theoretic research, although whether it will ever result in a change of the underlying axioms is highly dubious (in my opinion).

*Filling in holes. This is the closest analogy with the reals case. The idea is: if some set-theoretic object can be "well-approximated" by objects which our current axioms can prove exist, then we should be able to prove that that object exists itself. While large cardinals themselves can be viewed as a goal of this philosophy, I think it's maybe not the most exact instance available to us. The forcing axioms provide a robust example of this philosophy: basically, they say that the set-theoretic universe contains "highly generic" objects. Martin's Maximum can be viewed as a natural stopping point of this line of inquiry. Interestingly, forcing axioms and large cardinals are deeply intertwined.

I have to leave my computer unfortunately, but I'll add to this answer later tonight.
A: For category theory, sometimes you could need to handle the set of all sets and this set does not exist in $ZF$. But using $NBG$, you can speak of the class of all sets which would be similar to the set of all sets but untouched by the same problems that arise when you speak about the same concept in $ZF$. So $NBG$ possess a larger ontology than $ZF$ and indeed, $ZF$ can be constructed as a subsystem of $NBG$. I'm just not sure if this is the kind of completing you're looking for, but if this is, take a look at the first chapter of Goldblatt's: The categorial analysis of logic.
I guess that these theories can be completed by generating a new theory in which certain sets that were not accepted in the previous theories exist for the new ones, just as the set I mentioned before. Take $ZF$ and remove the axiom of the power set, it will be a less complete theory, that is: You won't be able to construct certain sets that could be built with $ZF$. 
This reminds me of an exercise in the beginning or Jech/Hrbáček's: Introduction to Set Theory They give the axioms of existence, extensionality and comprehension. A little later in the book, they ask you to prove that with these axioms alone, you can only construct $\emptyset$.
Start reading what I suggested, soon Asaf will emerge from the darkness and settle this down.
A: Going from the rationals to the reals makes the first-order theory of the ordered algebraic structures weaker: the first-order theory of the rationals is not decidable but the first-order theory of the reals is. You are confusing the logical strength of a theory with its apparent ontological requirements: in the example you give, the theory of $\Bbb{Z}[\sqrt{2}]$ can easily be interpreted in the theory of $\Bbb{Z}$. It is wrong to say that the theory of $\Bbb{Z}[\sqrt{2}]$ is stronger than the theory of $\Bbb{Z}$.
A: Feferman has done work in this area under the rubric of "unfolding" formal systems.
A: I think what you're looking for are reflection principles. A reflection principle asserts that there is a set that looks like (in some precise way) the universe of all sets.
Take your analogy of the real numbers $\mathbb{R}$ being the metric completion of the rational numbers $\mathbb{Q}$. It is natural to think that if a sequence in $\mathbb{Q}$ appears to converge in the metric, then perhaps there is a larger space, containing (an isomorphic copy of) $\mathbb{Q}$, in which it actually does converge. There is such a space, namely $\mathbb{R}$.
Now analogously, say we have a sequence of axioms $a_1, a_2, \ldots a_n$ for set theory. If these axioms appear to be consistent, then perhaps they are. If they are, this means that the statement $a_{n + 1} = $ "Axioms $a_1, a_2, \ldots, a_n$ are consistent" is true, so we can add $a_{n + 1}$ as a new axiom, producing a stronger set theory. Furthermore, it's an obvious candidate for a new axiom, because we know it cannot be proven from the previous axioms (given they can encode basic arithmetic) by Godel's second incompleteness theorem.
This approach is one example of a reflection principle, since by Godel’s completeness theorem, a set of axioms is consistent if and only if it has a model. That is, there is a set in which the axioms hold. So we see that in set theory, there is at least one very obvious way to extend a given universe of sets $V$ to a larger, more complete universe. Namely, consider a universe in which $V$ itself is an actual set! We don’t know all the properties that should hold in $V$, all we have is our specific set of axioms. But we can choose some or all of those axioms, and add an axiom saying that a set exists that satisfies them.
For example, the set $V_\omega$ of all hereditarily finite sets is a model of all the axioms of ZFC except the Axiom of Infinity. For instance, it satisfies the Axiom of Power Set, because if $x \in V_\omega$ then $\mathcal{P}(x) \in V_\omega$. This means that if took we took ZFC but left out the Axiom of Infinity, then our universe of sets would look like $V_\omega$, even though $V_\omega$ would not be a set. Then if we added in the reflection principle "There is a set that looks like the universe, in the sense that it is non-empty, transitive, and closed under power set" as an axiom, we would obtain the actual set $V_\omega$ (in general we would obtain a superset of $V_\omega$ but could then recover $V_\omega$ as a subset).
