My newest favourite number is 48 or more accurately arccos (2/3). Other favourite numbers are 5,17, and 309. Although 48 is ten more that the Revolting Cocks song 38, the reason for 48 degrees is explained (mathematically) below, but essentially 48 degrees is the solution to the problem:
A smooth hemisphere with radius, r, is resting in a fixed position on a horizontal plane with its flat face in contact with the plane. A particle of mass, m, is slightly disturbed from rest at the highest point of the hemisphere. Find the angle to the vertical that the particle leaves the surface of the sphere.
Also this is independent of radius, particle mass and local gravity! Nice.