$\displaystyle \pi$ provides a good example of what I mean with regard to economy of rounding. 3.14 is often the value used, because the 159 that follows is not seen as altering the value of the preceding 4 that much. However, I prefer to use the value 3.1416, and that seems a good, practical place to end it, because the 6 is a close rounding of the 59265 that occurs at that decimal place. Now, suppose the value of pi were 3.1449.... In that case, 3.14 would be a less useful approximation, and most likely, 3.145 would be more commonly used.
And that brings me full circle back to the title of this thread, "A good place to end pi." For most practical purposes, 3.1416 seems a good place to end it, providing more than enough precision. Of course, close-tolerance mechanical, engineering, or scientific work may require more precise values of $\displaystyle \pi$.
Suppose we have a real number $\displaystyle \displaystyle a$ which we round $\displaystyle \displaystyle n$ digits after the decimal point, obtaining an approximation $\displaystyle \displaystyle b$. Then you are expressing interest in finding small values of $\displaystyle |a-b| \cdot 10^n$. Contrast with the absolute error which is just $\displaystyle |a-b|$.
Edit: Another thing I thought of; I see the usefulness of writing 1.5 in place of 1.4999999 as it allows us to write the approximation 1.4999999 $\displaystyle \approx$ 1.500000 with fewer characters (to save our poor fingers from all the typing/writing they do), but consider that in working with significant figures and scientific notation we will often want to specify the precision, so we might end up writing 1.500000 anyway.
If I understand you correctly, all that mathematicians have been able to obtain is an approximation of the true value of $\displaystyle \pi$. Is that because we cannot use current methods to calculate $\displaystyle \pi$ out to an infinite number of decimal places, or is it because the methods yield results that cannot be replied upon to give correct digits all the way to infinity? In other words, when I read that $\displaystyle \pi$ has been calculated to a few hundred million decimal places, are those hundred million digits assumed to be correct, as far as it goes? Or is it that an error may creep in long before the calculation is stopped?
No! We can calculate $\displaystyle \pi$ to whatever decimal or any other precision we like, but because $\displaystyle \pi$ is transcendental no finite rational representation will represent it exactly.
Strange as it may appear being just morals we are restricted to finite representations. But the above does not mean that we have no finite representations that represents $\displaystyle \pi$ exactly. A transcendental number is an equivalence class of sequences of rational numbers and any element of an equivalence class may be used for that number. We have a Turning machine which given an integer $\displaystyle N$ will return the $\displaystyle N$-th decimal digit of $\displaystyle \pi$. This Turing machine characterises $\displaystyle \pi$ exactly, it is (exactly) $\displaystyle \pi$ in most senses that make sense.
(note I am avoiding restricting myself to spigot algorithms, so though the known spigot algorithm for $\displaystyle \pi$ is in base 16 that is not relevant here, but it to would do as an exact representation of $\displaystyle \pi$)
CB
Regarding the whole physics/math/approximation/granularity of the universe discussion: there is a well-known joking quote, "An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn't care." I mention it because I believe both directions of approximation have been mentioned in this thread.
As for practicality of having so many digits of pi... it seems there is at least one use: quoting from a Wikipedia article, "These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers." Granted, I don't feel this quote is entirely reliable in terms of how it's written and its belonging to a freely editable encyclopedia, but I'm guessing it's true that calculating digits of pi is a reasonable test for the performance of supercomputers.