I don’t have anything original here, but it’s fun. So, xkcd has a comic about interesting approximations and includes this one: (7^(e-1/e)-9)π². To keep the surprise a bit, try it and I’ll put more after the fold.
This is the number from this song:
What’s fun (from here and here) is that it (8675309) is also a twin prime (it’s prime and so is 8675311). Since it’s of the form 4n+1, it’s also the sum of two squares (422 and 2915) and is part of a primitive Pythagorean triple (8,675,309²=2,460,260²+8,319,141²). The fact it’s a Pythagorean triple follows from the fact that it’s the sum of two squares:
if a=b²+c² with b>c, then a²=(b²-c²)²+(2bc)²