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.
The answer is no, there is no exponential bound on PR.
PR contains Knuth's up-arrow functions, Elementary functions, etc.
PR is equal to the union of Grzegorczyk hierarchy.
Exponential functions appear at the third level of the Grzegorczyk hierarchy.

PR can alternatively be defined using the iteration function in place of recursion.

A good reference for information about these classes is the first chapter of second volume of "Classical Recursion Theory".

Also have the general function growth hierarchies like 
Grzegorczyk hierarchy,
slow-growing hierarchy, and
fast-growing hierarchy.