An amazing pandigital approximation to that is correct to 18457734525360901453873570 decimal digits is given by
(1)

found by R. Sabey in 2004 (Friedman 2004).
Castellanos (1988ab) gives several curious approximations to ,
(2)
 
(3)
 
(4)
 
(5)
 
(6)
 
(7)

which are good to 6, 7, 9, 10, 12, and 15 digits respectively.
E. Pegg Jr. (pers. comm., Mar. 2, 2002), found
(8)

which is good to 7 digits.
J. Lafont (pers. comm., MAy 16, 2008) found
(9)

where is a harmonic number, which is good to 7 digits.
N. Davidson (pers. comm., Sept. 7, 2004) found
(10)

which is good to 6 digits.
D. Barron noticed the curious approximation
(11)

where is Catalan's constant and is the EulerMascheroni constant, which however, is only good to 3 digits.