Pi appears a LOT in trigonometry, but only because of its 'circle-significance'. Does pi ever matter in things not concerned with circles? Is its only claim to fame the fact that its irrational and an important ratio?
[Math] Does the number pi have any significance besides being the ratio of a circle’s diameter to its circumference
pi
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"This is fairly trivial if we approximate the circles by regular $n$-gons and take the limit as n goes to infinity." You describe one of the most sophisticated techniques in ancient Greek mathematics. It was most assuredly not fairly trivial.
Study the Elements closer and you will learn Euclid's style. Ancient Greek geometry had a completely different view of mathematics, from the foundations up. Any of the ancient Greeks would find the modern notion of a limit absurd. They would reject it on the grounds of a completed infinity, and it would never appear in any of their polished works. I'm not saying I side with them; but that's what they'd say.
The answer, though anticlimactic, is simple: The question you ask is totally alien to the language and spirit of the Elements. In Euclid's work, $\pi$ does not yet appear. Sure you can tease it out using modern interpretations. But overall, Euclid is far more interested in the special ratios involved in constructing structures, rather than describing them.
Studying the history of mathematics, the unprepared student is particularly prone to hindsight bias. Sadly so are many of the popular expositors of the subject!
Turns out the answer is "yes". If you have a spherical geometry, it is easy to find an example: Draw a circle around the equator of the sphere. Now draw a line across the diameter of that circle -- it will arc up through one of the poles and back down on the other side. The length of that arc is obviously exactly half the distance around the sphere, so the ratio of circumference to diameter will be 2, which is a rational number.
Outside of Euclidean geometry, different circles can have different ratios in the same geometry. There is no singular "circumference to diameter ratio".
Best Answer
It is difficult to know if a circle is not lurking somewhere, whenever there is $\pi$, but the values of the Riemann zeta function at the positive even integers have a lot to do with powers of $\pi$: see here for the values.
For instance, you can prove that the probability that two "randomly chosen" positive integers are coprime is $\frac{1}{\zeta(2)} = \frac{6}{\pi^2}$.