1      2       4        8          16         
   S = --- + ----- + ----- + ------- + ---------- + ..... 
        2      5       41     3,281    21,523,361              

Almost there (not a proof):

Analyzing values for 1-S(n) it can be seen that

1-S(n) = 2^(n+1) / (3^(2^(n+1))-1)

and the sum approaches 1.  Now if I could only relate that somehow to the terms of S.  The equations are, not surprisingly, very similar.