In principle, not every number can be directly expressed as the result of a nontrivial Ackermann function or Knuth operator (just as not every number is a squar

In principle, not every number can be directly expressed as the result of a nontrivial Ackermann function or Knuth operator (just as not every number is a square or a cube) - although there is a section in the Knuth page you linked on how to represent any nonnegative integer as a series of Knuth-type operations. I would imagine, though, that to represent an arbitrary integer between 0 and n still needs ceil(log_2(n+1)) bits - since you have to distinguish between n+1 choices, whichever notation you use.