This is the Euler-Mascheroni constant, denoted by the Greek lowercase letter gamma (γ).
The harmonic series is defined as 1 + 1/2 + 1/3 + 1/4 + ... + 1/n and it diverges to infinity as n approaches infinity. As n approaches infinity, the difference between the sum and ln(n) approaches γ.
It is not proven that γ is transcendental. In fact, it's not even proven that γ is irrational, even though it most likely is!