Others have detailed the sequences that converge to pi or fractions of pi. On reason pi is so darn useful is because it helps measure the unit circle (circle of radius 1). The unit circle is kind of like the prototype cycle–and nature loves cycles. In fact, one of the reasons I bailed from my physics major was because I got tired of every course coming back to oscillations every time.
As for e, it’s a useful number because it’s the solution of the equation e[sup]x[/sup] = d/dx(e[sup]x[/sup])–it is its own derivative. In other words, if you graph y=e[sup]x[/sup], you get a function where the value y at any point x is the same as the slope of the function–the rate of change is proportional to how big it is. This is how many things in nature work, like the growth of bacteria in a plentiful food source (each individual bacterium reproduces at the same or similar rate, hence the rate of reproduction of the colony depends on how big the colony is). Also, radioactive decay follows this model, as each atom of a radioactive isotope has the same chance of decaying, the number of atoms that decay in a given time period depend on how many atoms there are at the time.
i was defined to solve the equation: x[sup]2[/sup]+1=0. Hence i is defined such that i[sup]2[/sup]=-1. This was important in algebra (for instance) because it solves that equation, and it means that a polynomial (equations of the type: x[sup]3[/sup]+x[sup]2[/sup]+x+1=0) of degree (highest power) of ‘n’ has ‘n’ values which satisfy the equation.
Note that powers of i have interesting properties.
i[sup]1[/sup] = i
i[sup]2[/sup] = -1
i[sup]3[/sup] = -i
i[sup]4[/sup] = 1
i[sup]5[/sup] = i
Indeed, multiplying imaginary numbers results in cyclic sequences like the above. Remember how pi is useful in describing cycles? Well it turns out that with a little effort, it’s possible to show that:
e[sup]i*x[/sup]=cos(x) + i sin(x) (Euler formula)
and that
e[sup]i*pi[/sup] + 1 = 0 (which relates all the important constants in one equation!)
The classical name for these things is “Mathematical Monsters.” 
Examples are the Sierpinski Gasket or Sierpinski Sieve, the Sierpinski Carpet, and the Menger Sponge. (The Koch Snowflake has already been mentioned.) The Menger Sponge for instance has infinite surface area but zero volume.
The neatest (IMO) example of this type is called the Cantor Set. It works like this: take the closed interval 0 to 1 (written [0,1], which means it includes all the numbers between 0 and 1, as well as 0 and 1 themselves. The open interval which doesn’t include 0 and 1 is written (0,1)). Now remove the middle third of the set, the open interval (1/3, 2/3) (which doesn’t include 1/3 and 2/3, but does include the numbers between). Now we have two parts left, [0, 1/3] and [2/3, 1]. Take each of them and remove the middle third, ad infinitum. In the end, how many points are left? Answer: the same number as you started with! Or, that is, for each point in the interval [0,1], you can find a corresponding point in the Cantor Set. Pretty funky.
Anyway, my favorite “monster” is derived from the graph of y=1/x. (You can graph it at a site like http://people.hofstra.edu/staff/steven_r_costenoble/Graf/Graf.html). The natural logarithm of a number (say ‘z’) is defined as the integral from 1 to z of the function 1/x. The natural logarithm (spelled ‘log’ or ‘ln’) is the inverse of the e[sup]x[/sup] function. Hence ‘ln(e[sup]x[/sup])=x’. Anyway, spin the graph around the x axis, and you get a kind of funnel-like structure. If you evaluate the funnel, you find that the funnel has finite volume but infinte surface area. Which means you can fill it with paint, but you can’t paint it!
Math is neat!