1  x  x^2 x^3  x^4   x^5    x^6     x^7
  ---------------------------------------------------------------
  1  0  0    0    0     0       0         0      for f0(x)=1
  1  1  0    0    0     0       0         0      for f1(x)=(x+1)^f0(x)
  1  1  1  1/2  1/3  1/12    3/40    -1/120      for f2(x)=(x+1)^f1(x)
  1  1  1  3/2  4/3   3/2   53/40   233/180
  1  1  1  3/2  7/3     3  163/40  1861/360
  1  1  1  3/2  7/3     4  243/40  3421/360
  1  1  1  3/2  7/3     4  283/40  4321/360
  1  1  1  3/2  7/3     4  283/40  4681/360
  1  1  1  3/2  7/3     4  283/40  4681/360

1  x  x^2 x^3  x^4   x^5    x^6     x^7    for
  ---------------------------------------------------------------
  1  0  0    0    0      0     0        0
  .  1  0    0    0      0     0        0    (x+1)-1
  .  .  1  1/2  1/3   1/12  3/40   -1/120    (x+1)^(x+1)-(x+1)
  .  .  .    1    1  17/12   5/4  469/360    (x+1)^(x+1)^(x+1)-(x+1)^(x+1)
  .  .  .    .    1    3/2  11/4     31/8
  .  .  .    .    .      1     2     13/3
  .  .  .    .    .      .     1      5/2
  .  .  .    .    .      .     .        1
  .  .  .    .    .      .     .        0

exp(-lambertw(-log(1+x)))-1    \ Pari/GP gives the limiting set of
                                     \ coefficients for the infinite iteration height
      %20 = x + x^2 + 3/2*x^3 + 7/3*x^4 + 4*x^5 + 283/40*x^6 + 4681/360*x^7 + O(x^8)