What is the fastest growing primitive-recursive-function?

Question

Fast growing functions tend to be not primitive-recursive. So I wonder if there is a limit how fast a function can grow, if it is known that it is primitive recursive.

• What is the fastest growing primitive-recursive function ?

A function $f$ is said to grow faster than a function $g$, if $f$ eventually surpasses $g$ no matter how big the head start, which is given to $g$, is. In other words, for every natural number $k$ there is natural number $n_0$, such that $f(n)>g(n+k)$ for all $n\geqslant n_0$.

Answer 1

Using your definition of "grows faster" in the comments, for any primitive recursive increasing function $f(n)$, $f(2n)$ grows faster. (or $f(n^2)$, $f(2^n)$, etc.)

If you want much faster, take any primitive recursive function $f$; it will be slower growing than $g(n) = 2 \uparrow^m n$ for some $m$. So increase $m$ greatly; for example, $h(n) = 2 \uparrow^{m \uparrow^m m} n$ will be much faster growing than $g$, and therefore $f$.

Answer 2

If $f(n)$ is primitively recursive, then so is $f(n)+1$, which, under any reasonable definition I can think of, grows faster than $f(n)$.