User blog comment:Hyp cos/tree function and TREE(3)/@comment-5150073-20140624084600/@comment-25418284-20140624204753

My friend, $$n \mapsto \omega_1^\text{CK}[n]$$ is most certainly not a computable function! For some small values of n, we can prove values of $$\omega_1^\text{CK}[n]$$, but as n increases, the amount of time/memory we'll need approaches infinity. It's just like BB(n) in that it quickly becomes extremely difficult to compute.

Of course you can write down $$f_{\omega_1^\text{CK}[65536]}(m)$$, but the problem arises when we try to evaluate it.