User blog comment:LittlePeng9/Long hierarchies of functions (on my other blog)/@comment-11227630-20171101003439

What about using other definition of "eventually outgrow"? Such as Sbiis Saibian's definition:
 * $$f[<]g$$ if for all m and k, there exists N such that for all $$n>N,\ f(n+m)+k 0, $$f(an+b)>g(n)$$.

Then "f strictly eventually outgrows g" if $$f\ge^*g$$ but $$g\ge^*f$$ doesn't hold.

These definitions allow the n inside change linearly, so they are stronger than your "eventually domination" and "properly domination".
 * For example, $$2^n+n^2$$ properly dominates $$2^n$$ in your definition, but it doesn't eventually outgrow $$2^n$$ in Sbiis Saibian's definition or in my definition.

Then how long can an eventually domination chain be if we use these stronger conditions?