The classical (in true historical sense) application is the sand reckoner and the cattle problem (it might be that Archimedes used problems with practically incalculable numbers to discover who among the mathematicians in Alexandria claimed as their own results that Archimedes sent to Alexandria without proof). These numbers were as big for the time as the numbers you cite are today. Note also that very rapidly increasing integer sequences are related to undecidability results, see the work of Harvey Friedmann and others
(Provocative side note: for what we known, it might be that the "separation" that we experiment between NP complete problems and P problems might due to the fact that P might be equal to NP but with polynomials with incredibly large degree and coefficients)
Edit: I agree with @Per Alexandersson that it would be better to give examples where the specific number (and not only its vague hugeness) matters. In fact we can say that "vague hugeness" examples are uninteresting precisely because the sand reckoner for the first time showed how easy it is, in principle, to give such examples (before that someone even doubted that big numbers can be specified, or even that the existing sand was finite in number; limit case: some primitive languages count "one, two, many").
In my opinion the cattle problem is a case where the specific number matters:  it asks the specific number;  it was not obvious (at the time, well before the Indian algorithm) that a solution exists at all (it is not a problem of "finite vs infinite");  it is not obvious that if a solution exists it must be quite big. The function that associates to a squarefree $N$ the least natural $x$ such that $Nx^2$ has distance 1 from a square might have looked almost completely random at the time; it is not a isotone function (Vardi specifies smaller data that would give much bigger numbers) which in a way makes it even more interesting, despite its relative smallness (but not for its time), in comparison with the rapidly increasing sequences that also Timothy Chow and Todd Trimble consider logically interesting and with specific combinatorial meanings for the specific values.
As @Pietro Majer spotted, the parenthetical remark about Archimedes is only speculation (that I found on the net about a month ago, and that now I cannot trace back. The source did not give real substance to the claim, and so i did not bookmark the url). However, I cited that since in any case it is a plausible application of "too-big" numbers. After all, in the same way as Archimeds could have done with plagiarizing mathematicians ("specify the number please"), Pietro Majer invited me to sustantiate a probably unsustantiable claim.
The following might be a reasoning behind this speculation (for a amusing possibility, in the same way as I cite a possibility concerning P and NP).
In the preface to On spirals Archimedes relates an amusing story regarding his friends in Alexandria. He tells us that he was in the habit of sending them statements of his latest theorems, but without giving proofs. Apparently some of the mathematicians there had claimed the results as their own so Archimedes says that on the last occasion when he sent them theorems he included two which were false:
... so that those who claim to discover everything, but produce no proofs of the same, may be confuted as having pretended to discover the impossible.
However, "false" is not the only translation: from pag. 20 in http://www.ifi.unicamp.br/~assis/Archimedes-2nd-edition.pdf
there are two included among them which are impossible of realisation [and which may serve as a warning] how those who claim to discover everything but produce no proofs of the same may be confuted as having actually pretended to discover the impossible.
A footnote by Heath on "impossible of realisation" notes a corruption of the Greek text and concludes
The meaning appears to be simply "wrong".
But if one takes "impossible of realisation" also the cattle problem (and not only false propositions) would fit.
However, this is not a proof, and even if it were a proof one can cite
what a historian said to Kolmogorov
You have supplied one proof of your thesis, and in the mathematics that you study this would perhaps suffice, but we historians prefer to have at least ten proofs.
So you are justified in reacting to the speculation I cited with (the much more classicaly supposed last words) "Noli turbare circulos meos" (or their equivalent in vernacular contemporary italian, increasing by 1 the dimension).