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I'll be giving a talk at Albion College on October 3rd, on "Fast growing functions".
It is meant to be accessible to undergrads with no prior knowledge of mathematical logic, although it will touch on some results on provability.
@AndresCaicedo The talk sounds great. It should be clear to me which results are being described in the abstract, but it is not 
.
The results that come to mind are those of ordinal analysis characterizing the provably-recursive functions of a theory, but I'm not sure which classical principles can be understood in this context.
@Convention_T Thank you!
The talk is aimed at undergrads, so sadly I won't be mentioning ordinal analysis.
And maybe "classical principles" is grossly inaccurate. 
I just meant that there are no new examples in the talk, it is all known results (at least, known among logicians) of a combinatorial rather than 'logical' character. For instance, I mean to talk of Friedman's function from his "long finite sequences" paper.
@AndresCaicedo Oh haha, thanks for clarifying!