When would you need a lot of decimal places for exact measurement

When would one need a number as fine/precise such as 1.00000000000000000000000000001 or much more? Micro physics? Like, when dealing with atomic or sub atomic exact measurements? that's my guess but I'm no genius so I have no idea to be honest. Any applied math experts out there willing to shed some light on this for me? Again, in what applied field would one need a very exact number such as pi solved to the 10 (to the ninth power) decimal place.

If you could provide a research link or something that would be great too! thanks :)

If you calculate π to 100 places (entirely possible) you will not get a more accurate estimate of the
circumference of a circle the diameter of which you took a guess at.

Adding places after the decimal means nothing unless the figures are significant.
(See: significant figures, meta-data)

High precision is used in some applications, GPS is a good example.

I'm no maths expert, but maybe space exploration needs to be this accurate. When calculating the distance to remote galaxies, doppler shift from light from these galaxies, things like that.
If you consider that all the radio energy so far collected from space for research, is less than the heat in one cigarette end, then you must be in the realm of working to such small figures.
Hope this helps.