Does the number pi have any significance besides being the ratio of a circle's diameter to its circumference? 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?
 A: 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 $\frac{1}{\zeta(2)} = \frac{6}{\pi^2}$.
A: $\pi$ appears in Stirling's approximation, which is not obviously related to circles.  This means that $\pi$ appears in asymptotics related to binomial coefficients, such as
$$\displaystyle {2n \choose n} \approx \frac{4^n}{\sqrt{\pi n}}.$$
In other words, the probability of flipping exactly $n$ heads and $n$ tails after flipping a coin $2n$ times is about $\frac{1}{\sqrt{\pi n}}$.  This asymptotic also suggests that on average you should flip between $n + \sqrt{\pi n}$ and $n - \sqrt{\pi n}$ heads. 
A: Because of the formula $e^{i\pi}+1=0$ you will find $\pi$ appearing in lots of places where it's not clear there is a circle, e.g in the normal distribution formulae.
A: Yes, the ratio $\pi$ of a circle's circumference to its diameter shows up in many, many places where one might not expect it!
One partial explanation (similar in spirit to "circles lurk everywhere") is that the equation for a circle is a $quadratic$ (eg. $x^2+y^2 = r^2$.)  After nice linear functions, the next most commonly used functions are quadratic functions and everywhere one runs into a quadratic function, a trig substitution (e.g. $x = r \cos \theta; y=r\sin \theta$) may be useful, turning the quadratic function into something involving $\pi.$  This explains the antiderivative $\int \frac{1}{1+x^2} dx$ involving $\pi$, the sum of reciprocals of squares $\sum^\infty\frac{1}{k^2}$ involving $\pi$ and the area under the Gaussian distribution involving $\pi$.  And so on....
A: I am late to the party, but I found a curious appearance of the decimal expansion of $\pi$ in the lecture notes for the Dynamics and Relativity course by David Tong, in section 5.2.2.
We have a one-dimensional problem with two bouncing balls of mass $M$ and $m$ which can collide elastically. The leftmost ball has mass $M = 16 \times 100^Nm$, the smaller ball is on the right of the large ball. There is a perfect wall on the right of both balls (the wall does not move upon collision). $N$ is an arbitrary integer.
If a push to the right is given to the ball with mass $M$, it will move toward the smaller ball and upon collision, the smaller ball will start moving towards the wall, until it bounces back toward the larger ball. They will again collide, taking away some of the momentum of the large ball and allowing the smaller ball to move toward the wall again.
After a finite number of collisions, the large ball will start moving in the left direction. The number $p(N)$ of collisions of the balls $M$ and $m$ required to reach this situation is the first $N+1$ digits of $\pi$ (so $p(0) = 3$, $p(1) = 31$, $p(2) = 314$, $p(3) = 3141$, etc). (More precisely, it is given by $p(N) = \lfloor 10^N \pi \rfloor$).
For further details, I invite you to read the lecture notes by David Tong linked above. Basically there is a matrix recurrence relation for the velocities of the balls after each collision, and the solution to this recurrence equation involves the $\sin$ and $\cos$ functions.
