Why does the author subtract the minimum $\ln\,\ln([1-F(t)]^{-1}])$ for the Weibull plot? When I read the Weibull Analysis Handbook (Abernethy et al., (1)), in Table 2.1 (p15), I don't really know how he gets the right-hand column ("Col 2 value - Min Col 2 value (-6,91)). Does he assume it or take it from a formula?

Earlier on the same page he gives the formulas

(1)  Abernethy, R.B., Breneman, J.E., Medlin, C.H., and Reinman, G.L. (1983),
Weibull Analysis Handbook,
 Pratt & Whitney Aircraft, Government Products Division, United
Technologies,
P.O. Box 2691, West Palm Beach, Florida 33402
 A: The document describes the construction of a form of Weibull probability plot (which back 33 years ago was commonly done on Weibull plotting paper, which supplied conveniently scaled axis and gridlines to save calculations like $\ln(\ln[(1-F)^{-1}])$, since one could simply work with percentile-ranks -- $F$ or $1-F$ -- directly).
In the process of discussing plotting by hand, it's necessary to compute an abscissa and an ordinate (i.e. actual x and y plotting positions). These may be shifted for convenience without altering the appearance of the plot (and without changing what tick mark labels are applied to a point).
As a result, since $\ln(\ln[(1-F)^{-1}])$ is largely negative, for convenience the ordinates are shifted by their minimum value. This makes it easier to construct a plot. It is neither based on a particular formula (one could make one but it's rather an obvious calculation, so seen as unnecessary), nor is it based on an assumption -- it is simply presenting the calculation of a convenient plotting position. 
If you have done a lot of hand-plotting, such scaling calculations (and other forms of normalization) are quite natural.
