n (n Knuth up arrows) n (n Knuth up arrows) n = n (n+1 Knuth up arrows) 3, so that's not really impressive either. Your function is comparable in size to f_omeg

n (n Knuth up arrows) n (n Knuth up arrows) n = n (n+1 Knuth up arrows) 3, so that's not really impressive either. Your function is comparable in size to f_omega(n) on the fast-growing hierarchy. [1] At any rate, it's known that BB(18) is larger than Graham's number [2]. In fact, the same person has a bunch of long-lasting Turing machines for values around 18 [3], giving the lower bounds: BB(15) > f_omega(2046) > 2046 (2046 up arrows) 2046 BB(16) > f_omega(f_omega(5)) BB(17) > f_(omega+1)(10) BB(18) > f_(omega+1)(7e57) > Graham's number BB(19) > f_(omega+1)^3(9) > f_(omega+2)(3) BB(20) > f_(omega+1)^4(7e57) > f_(omega+2)(4) BB(21) > f_(omega+2)(f_(omega+1)^2(4)) BB(22) > f_(omega+2)^3(f_(omega+1)^2(4)) And for larger values again [2]: BB(38) > f_(omega2)(167) BB(64) > f_(omega^2)(4098) BB(85) > f_(epsilon_0)(1907) Of course, we expect the Busy Beaver function to grow a lot faster than these bounds. [1] https://en.wikipedia.org/wiki/Fast-growing_hierarchy https://en.wikipedia.org/wiki/Fast-growing_hierarchy [2] https://googology.wikia.org/wiki/Busy_beaver_function#Graham.27s_number_vs._.CE.A3.28n.29 https://googology.wikia.org/wiki/Busy_beaver_function#Graham... [3] https://googology.wikia.org/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham's_Number https://googology.wikia.org/wiki/User_blog:Wythagoras/The_ni...!