Mostly, for the functions in the fast growing hierachy, LOWER bounds are given like
$f_k(n) > 2 \uparrow^{k-1} n$
but what abour (reasonable tight) UPPER bounds ?
What are the best known UPPER bounds for $f_k(n)$ ?
The functions in the fast growing hierachy are defined as follows
$f_0(n) = n+1$
$f_{k+1} = f_k^n(n) = f_k(f_k(...n)...)$
where $f_k$ appears n times.
So $$f_1(n) = 2n$$ $$f_2(n) = n2^n$$ $$f_3(n) \ge 2 \uparrow 2 \uparrow ... \uparrow 2 \uparrow n$$
with n 2's
and so on.
