User blog comment:VoidSansXD/Possibly non-recursive function/@comment-37246647-20190218235317/@comment-30754445-20190219005515

That's right. We can never have complete information on the values of a nonrecursive function.

We may be able, however, to get partial information. For example, we sometimes can:

(1) Use case-by-case reasoning to find the value of the function for certain small values of n.

(2) Provide LOWER BOUNDS of the function for specific larger values of n.

(3) Provide FORUMLAS for lower bounds of the function that are correct for all n.

For example, any proficient C programmer could easily show that you can get an output of n^^n with 100+10n characters. So f(100+10n) > n^^n for all n.

You can get better bounds by being more clever with your programming. We actually had a competition here, a few years ago, to see who can give the best lower bound of f(512).

The winning entry (Loader's Number) is one of the biggest computable numbers known. In fact, it is so big that currently no human understands how to build it with ordinals (IOW nobody currently knows how to "count" all the way up to the ordinal that's needed to build Loader's Number)

And of-course this is still just a lower bound. The actual value of f(512) is probably much much bigger that.