%N Levels where minus signs appear in the infinite power tower representation of 2 = exp(exp(-exp(-exp(exp(-exp(exp(exp(-exp(-exp(-exp(...))))))))))).
%C Start with x=2, then at every step write down the step number if x is negative, replace x with log(abs(x)), repeat.
%C 2 = exp(exp(-exp(-exp(exp(-exp(exp(exp(-exp(-exp(-exp(...))))))))))) in the sense that 2 is the limit of the sequence of power towers with increasing heights generated from initial segments of this sequence: exp(1), exp(exp(1)), exp(exp(-exp(1))), exp(exp(-exp(-exp(1)))), exp(exp(-exp(-exp(exp(1))))), ... (the initial seed value here is taken to be 1, but it can be any number as the limit does not depend on it).
%e a(1)=3 because the first minus sign appears at level 3.
%t levels = 120; Flatten[Position[Sign[NestList[Composition[Log, Abs], 2, levels]], -1]]
%A _Vladimir Reshetnikov_, Apr 26 2013