User blog comment:Ynought/Function iterator?/@comment-24725252-20190524172302

This function breaks down beyond F^^F = because F^^(F+1) = F^(F^^F), when you evalute it is F^(F^F^F...) which is no bigger than F^^F, or e0. The reason is that is called the fixed point trap, and it pervented BEAF past tetrational arrays to becoming well-defined.

Another less serious comment is you forgot to include F^ I<\sup>k when k is a limt (not in the form of a+1 for some a). Presumably F^ i<\sup>k[n] = F^ i<\sup>k(n). that rule might be assumed by intuition but leaving it out makes the function ill-defined.