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Also got me thinking about what fraction is most "pi efficient"? That is, what fraction is closest to pi compared to how many digits you must use to represent the fraction (or perhaps only digits in the numerator or denominator)?

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Seems they answered that question in TFA.

Using continued fractions in Mathematica. Maybe there's enough information in the article to say that this is enough to have answered the question. I don't know enough about all of the methods to say, personally. Also, I never outlined what was the best metric to evaluate how well accuracy compares to the number of digits needed.

The answer is already there. If you extend the sequence given in the article longer, the ratio of the number of digits in the rational representation to the decimal representation gets worse, not better, so you know you reached the inflection point early.

As others have said, the convergents of the simple continued fraction representation give the best rational approximation [1] relative to the size of the fraction, in the strongest relevant metric: that of absolute error bound. Proofs can be found in any book on Diophantine Approximation; the classic (and readable) reference is Khinchin's Continued Fractions book.

[1] https://en.wikipedia.org/wiki/Continued_fraction#Best_ration...

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