User blog comment:Simplicityaboveall/The Construction of Extremely Large Numbers/@comment-25601061-20160725202417/@comment-1605058-20160729125735

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You are right that in my humble opinion "the FGH" is uniquely determined, provided we have specified fundamental sequences for ordinals. In my dictionary, "the FGH" is the hierarchy of functions defined by $$f_0(n)=n+1,f_{\alpha+1}(n)=f_\alpha^n(n)$$ and for limit $$\alpha, f_\alpha(n)=f_{\alpha[n]}(n)$$. The hierarchy you defined has the first two rules different, which is why I'd not call it FGH.