User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-5150073-20130414231904

Here is my simple definition of the fundamental sequence to w_1^CK:

Let the complexity of ordinal is the last n such that f_alpha(n) > S(n,n,n,n), where S(n) is the max shifts function for n-dimensional, n-head, n-color and n-state busy beaver.

Now, let w_1^CK[n] is the largest ordinal with complexity n. If there is no ordinals with complexity n, then we take the largest ordinal with complexity (n-1) and add 1 to it (to make strict increasing fundamental sequence).

I was inspired to it when I read this post: http://forums.xkcd.com/viewtopic.php?f=17&t=79956.

That allows us to define not only f_w_1^CK(n), but also things like f_w_1^CK*2(n) = f_w_1^CK+w_1^CK(n), since we can take w_1^CK[n] now.