I found myself wondering the other day whether the fastgrowing functions from natural to naturals that are studied by people like proof theorists are the restriction to the naturals of analytic functions from the complexes to the complexes. It turns out, because of the Pringsheim Interpolation formula, that they all are. Now the location of the zeroes of the Riemann zeta function contains lots of information about primes. Might it be the case that the location of zeroes of some of these analytic continuations of fastgrowing functions contains information about countable ordinals? Is there an obvious reason why they can't? I do wonder why no connection between this material and complex analysis has ever been made....

These zeros can be literally anywhere, by the result you quote that for any discrete set $\{z_n\}$ and any values $a_n$, there are entire functions with $f(z_n)=a_n$. 


Here is an immediate reply to your question, cut and pasted from the Wikipedia article on the factorial function:
That is, there is no unique way to do what you want, even for the factorial function. There is not even a universally agreed upon best way, or agreement upon what "best" might mean. To add content to your question, I think a few examples are called for, and perhaps a sensible restriction on the allowed extensions. 

