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 [2] https://googology.wikia.org/wiki/Busy_beaver_function#Graham... [3] https://googology.wikia.org/wiki/User_blog:Wythagoras/The_ni...!