Can we hope to solve all Diophantine equations? According to Godel result, neither ZFC nor other particular theory is strong enough to resolve all questions about, say, Diophantine equations. But maybe we can hope that a sequence of theories will help? It is known that ZFC-1 theory (ZFC + Cons(ZFC) ) is much stronger than ZFC, in sense that now there are theorems with extremely shorter proofs, and many new theorem are now decidable. If we continue this to ZFC-2, .., ZFC-n, ... then ZFC-w which is union of all, ZFC-(w+1) and so on, we can continue to extremely large sets of theories, about all of them we have no doubts, and maybe now for every natural Diophantine equation we can choose a theory witch resolve it? Moreover, if we would be able to imagine non-enumerable set of such theories, may be we could hope that for EVERY Diophantine equation has a corresponding theory from this set in which it can be resolved? Or this is trivially incorrect “conjecture”? It seems that this does not contradict to Godel Theorem, which consider one theory, not a sequence of theories.
Another way of thinking about the same idea is to take only axioms from ZFC but add a new derivation rule, which would say that "from any set of axioms A it follows that A is consistent". With this derivation rule we would derive Cons(ZFC) in one step! So, for some reasons (by the way, I do not understand why) Godel theorem is not applicable here. May we hope that with ZFC extended with such a new derivation rule we can, say, solve all Diophantine equations? 
 A: On the one hand, no consistent theory $T$ that we can
describe by giving a computable enumeration of its axioms
can settle all Diophantine equations. The reason is that
for any such theory $T$, we can construct an integer
polynomial $p_T(\vec x)$ that has a solution in the
integers if and only if $T$ is inconsistent. Thus, since
$T$ is consistent, $p_T$ will have no solutions, but $T$
will not prove this.
One may construct $p_T$ by understanding the MRDP solution
to Hilbert's 10th problem. In that argument, for any Turing
machine program, one may construct a polynomial whose
solutions correspond to halting instances of the program.
But consider the program that searches for a proof of a
contradiction in $T$, halting only when one is found. The
corresponding polynomial $p_T$ for this program will have a
solution if and only if the program halts, which is if and
only if $T$ is inconsistent, as desired.
So we cannot hope to settle all Diophantine equations with
respect to one computably enumerable theory.
Nevertheless, we can describe theories that solve all
Diophantine equations in other ways. For example, the
theory TA known as True Arithmetic, extends PA and consists precisely of
the first order assertions that are true in the standard
model $\langle \mathbb{N},+,\cdot,0,1,\lt\rangle$. (One could use $\mathbb{Z}$ in place of $\mathbb{N}$ here, or just realize that $\mathbb{Z}$ is interpretable in $\mathbb{N}$.) Our
background theory ZFC proves that TA is consistent and
complete, and it certainly correctly settles all the
Diophantine equations, proving of exactly those polynomials
that have solutions that they do have solutions and of the others that they do not. So this
theory is the kind of limit theory you requested. But the
difficulty with TA and with any of the non-enumerable limit
theories that you seek, is that it is too difficult to
recognize the axioms. We cannot tell if a proof from TA
is legitimate, because we cannot even recognize the axioms.
For the purpose of settling Diophantine equations, it would
suffice to use $TA_1$, consisting just of the true
$\Pi_1$ assertions. But recognizing whether a given
$\Pi_1$ assertion is true or not (that is, recognizing it as an axiom of $TA_1$) is exactly as difficult
as recognizing whether a given Diophantine equation has solutions in the integers or not. 
This last difficulty is inherent in the problem, since any theory correctly proving whether the polynomials have solutions or not will prove all the instances of $TA_1$. So $TA_1$ a minimal instance of the theory you seek, but it is not clear how useful it is for your purpose, since it is as difficult to recognize the axioms of this theory as it is to solve the problems you intend to solve with it.
