Euler's Constant (or Mascheroni's constant) is the limit (as n goes to infinity) of
1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n - log n.
The Euler-Mascheroni constant is denoted by the greek letter gamma (γ) (or sometimes C) and has the numerical value approximately 0.57721566
Text for cut and paste:
Constant EulersConstant = 0.577215664901532860606512090082402431042
Here is a much longer string of digits for γ:
0.57721566 4901532860 6065120900 8240243104 2159335939 9235988057 6723488486 7726777664 6709369470 6329174674 9514631447 2498070824 8096050401 4486542836 2241739976 4492353625 3500333742 9373377376 7394279259 5258247094 9160087352 0394816567
Even though over 1,000,000 digits of this number have been calculated (http://dipmat.unife.it/Root/d-Mathematics/d-Number-theory/t-Eulers-constant-Haible), it is not yet known if it is a rational number (the ratio of two integers a/b). But if it is rational, the denominator (b) must have more than 244,663 digits!
Links to sites with a lot more detail
euler.htm - 22 September 1999, Jim Lux