User blog comment:Hyp cos/BEAF isn't so strong/@comment-1605058-20130921111214/@comment-5150073-20130921124304

If something doesn't defined formally, it doesn't means that it cannot be bounded just as natural continuation of well-defined. For example, what is 2.5^^^3? It is trivially the number between 2^^^3 = 65536 (a reasonably big number with can be even used for Earth purposes) and 3^^^3 (which is so large that physicists haven't found any applications for it)! So, maybe 2.5^^^3 is comparable to the number of stars in the observable Universe?

We just remember that hyper-operators are strictly increasing functions, and so the bounds can be worked out:

2.5^^^3 = 2.5^^(2.5^^2.5) > 2.5^^(2.5^^2) = 2.5^^(2.5^2.5) > 2.5^^9.882 > 2.5^^9

2.5^^^3 = 2.5^^(2.5^^2.5) < 2.5^^(2.5^^3) = 2.5^^(2.5^2.5^2.5) < 2.5^^(2.5^9.883) < 2.5^^8568