curve fitting with extremely “large” numbers

here is an unconventional but apparently workable idea. wondering if anyone has tried something like it, esp looking for references, examples, or nearby related work.

am working on a mathematical induction problem with a sequence $x_i$ of strings which are "generally increasing" in size as $i$ goes to infinity. from inspection the function is loosely related to the Fibonacci sequence.

these strings could be converted to very large multidigit base-N numbers (ie with something in the range of dozens of digits or more) and one could do a curve fit using standard algorithms eg Marquardt-Levenburg nonlinear least squares algorithm, powells algorithm, etc.

of course its preferable to get an exact curve fit such that it matches $x_i$ for some finite $i \leq k$ and then generalizes to unseen data $i>k$. ie induction.

so what are some of the largest magnitude numbers that curve fitting has been applied to, and esp cases where exact formulas have been obtained? are there some researchers, refs, or some software that specializes in this? etc

my question should not be taken to be asking about datasets with many datapoints. this question is a much different regime of interest where there may be "not so many datapoints", but each datapoint has a very large numerical magnitude with lots of digits. so its "big data" in a sense but with a key twist.

note however equivalently this question is the same as fitting floating point values to very high degrees of precision and the tight fit is not arbitrary but instead meaningful to the problem in some way.

(this question is partly inspired by the AMS ref on experimental mathematics.[1])

[1] Experimental Mathematics: Examples, Methods and Implications by Bailey/Borwein, notes of the AMS, 2005

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This question, in a more specific form, might get some great answers on the Mathematica site. For instance, it's possible that a single command like FindSequenceFunction[{1,1,2,3,5,8,13}] (with the values replaced by yours) would immediately solve your problem. Mathematica calculates symbolically and with arbitrary-precision and rational arithmetic; numbers of 100,000 digits will barely slow it down. Otherwise, I suggest posting it on the Math site, because exact fitting isn't really a statistical problem--it's a mathematical one. – whuber Dec 5 '12 at 16:09
ok thx yes, forgot to mention "generating functions" in the question. the data in my case seems to have a very complex generating function. will have to look into that mathematica cmd. – vzn Dec 5 '12 at 16:56