interesting pattern in hyper-operations

[BenStandeven]

would be approximately true by induction, and we always have e ^k^ 0 = 1, and e ^^ x = x+1 approximately on [-1, 0]. The degree of approximation might decay a bit at each level, I suppose.

Hm, you mean there is a function that is closest (say by maximum difference or by area between graphs) to the linear function on the interval [-1,0] among the solutions of f(x+1)=e[k]f(x)?

Yeah; I'm thinking that it will probably be e[4] or e[5], since for the higher functions I would expect the deviation from linearity to be greatest on the outer intervals like [-1, 0], and least near the middle.

What I rather meant is whether this would be a uniqueness criterion. I.e. whether there is only one function with minimal distance to the linear function [-1,0]? What do you think?

I doubt it; I'd think by composing with the right 1-periodic function, you can always move closer to linearity on that interval.