User blog comment:Revilo30703/Knuth's Up Arrow Infinite Expantion/@comment-27173506-20160123125539/@comment-27173506-20160124140403

I see the pattern (I think), but you have to put it in an actual ruleset if you want people to understand. Plus, I think using undefined tags would help you make things clearer. Here's a good ruleset (I think), where X means rest of expression:

1. a^X1=a

2. a^1,Xb=a^Xb

3. a^X,n+1,1,1,1...1,1b=a^X,n,a,a,a,a...a,ab (which could be strengthened by changing it to a^X,n,b,b,b,b...b,bb, as b is the one that grows).

4.a^X,n+1b+1=a^X,n(a^X,n+1b)

Now that I see this ruleset, I am sure that it reaches f_w^w(n).