Do you have an example of when 130 digits of calculator precision are needed?

For root finding algorithms in some edge cases it helped me decide the multiplicity of a root. Also, when calculating constants I wanted to make sure that the result was as precise as possible without any rounding errors introduced in the calculation.

Solving a "Ponder This" challenge like [1] (and getting the full bonus).

[1] https://research.ibm.com/haifa/ponderthis/challenges/August2...

Couldn't you simply use arbitrary precision floating arithmetic with i.e. Python?

Yes, python with import decimal can be used, as well as Linux bc, but the question was what you need so much precision for.

I've been nerd sniped.

What are the "rules" of these problems? Are you allowed to use calculators and programming? Or should the solutions be pencil and paper only Olympiad-style?

Yes, you can use all resources available to you. Most challenges involve some amount of programming.

> I've been nerd sniped.

Huh, looks like I'm one of today's lucky 10,000

https://xkcd.com/356/