Approximate the Fransén-Robinson constant
Mathematica, 44 39 36 25 UTF-8 bytes


-5 bytes thanks to Sp3000 
-3 bytes thanks to kennytm
-11 bytes thanks to senegrom


Crossed out 44 is still regular 44!!

N[∫1/x!{x,-1,∞},#+1]&amp;


Example:

f=N[∫1/x!{x,-1,∞},#+1]&amp;
f[2]


Outputs 2.81.

Explanation

N[               , # + 1] 
  ∫1/x!{x,-1,∞}


First step takes Numeric of the rest, with # (first parameter) + 1 precision. ! (factorial) does what you'd expect. {x, -1, Infinity} sets the bounds for the (strangely formatted) Integral.