# Growth rate of primitive recursive functions

Pardon me for asking a question whose answer is surely well-known, but is beyond my expertise:

Q. Is the growth rate of each primitive recursive function $f(n)$ bounded by some exponential $O(c^n)$, or $O(n^n)$, or ...?

I know Ackermann's function grows faster than exponential, but it is not primitive recursive.

• I am also not an expert but I am pretty sure that towers of exponents, and maybe even $2\uparrow^k n$ for any constant $k$ are primitive recursive en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation. – Sasho Nikolov Feb 6 '16 at 3:30
• Actually, after googling something on a hunch, it seems that for every primitive recursive function $f(n)$ there exists a constant $k$ such that $f(n) \le A(n,k)$, where $A$ the Ackermann function. – Sasho Nikolov Feb 6 '16 at 3:37