In other words: which physics experiment requires to know Pi with the highest precision?
Answer
Pi is very far from being the only number we need in physics. Typical theoretical predictions depend on many other measured and calculated (or both) numbers besides pi.
Nevertheless, it is true that one needs to substitute the right value of pi to get the right predictions. Therefore, the right answer to your question is the most accurately experimental verified theoretical prediction we have in physics as of today, namely the anomalous magnetic dipole moment of the electron
http://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment
In some natural units, the magnetic moment of the electron is expressed as a g-factor which is somewhat higher than two. Experimentally, $$\frac g2 = 1.00115965218111 \pm 0.00000000000074$$ Theoretically, $g/2$ may be written as $$\frac g2 = 1+\frac{\alpha}{2\pi} + \dots$$ where the $\alpha/2\pi$ first subleading term was obtained by Schwinger in 1948 and many other, smaller terms are known today. The theoretical prediction agrees with the experimental measurement within the tiny error margin; the theoretical uncertainty contains the effect of new species of virtual particles with the masses and couplings that have not yet been ruled out. This requires, among many and many other things, to substitute the correct value of $\pi$ in Schwinger's leading correction $\alpha/2\pi$. You need to know 9-10 decimal points of $\pi$ to make this correction right within the experimental error.
So in practice, $\pi\approx 3.141592654$ would be OK everywhere in the part of physics that is testable. However, theoretical physicists of course often need to make calculations more accurately if not analytically, to figure out what's really happening with the formulae.
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