Limit[(n! LaguerreL[n,-1]/(n^(n+1/4)/Sqrt[2]/E^(n-2 Sqrt[n]+1/2))-1) Sqrt[n], n->∞] 

Mathematica (wrong) output

13/16 = 0.8125 

The right result is:

31/48 = 0.645833... 

But numerically it is computed right (after ~ 1 hour):

N[Table[(n! LaguerreL[n,-1]/(n^(n+1/4)/Sqrt[2]/E^(n-2 Sqrt[n]+1/2))-1) Sqrt[n], {n, 1000000, 10000000, 1000000}], 20] 

{0.64595327485857865704, 0.64591815870538845793, 0.64590259799022250717, 

 0.64589332088298149047, 0.64588698941420000820, 0.64588231547803760347, 0.64587868275613942973, 0.64587575441366589107, 0.64587332876408306831, 0.64587127669377150813} 

I already reported this bug in 2012, but still was not fixed (in versions 7,8,9,10) http://code.google.com/p/mathematica/issues/list see Issue 46

For more please see my article "Too many errors around coefficient C1 in asymptotic of sequence A002720" http://members.chello.cz/kotesovec/math_articles/kotesovec_too_many_errors_A002720.pdf