User blog comment:Deedlit11/Is BEAF well-defined?/@comment-25418284-20121121205547/@comment-5150073-20121124094557

As I see, largest Bowers' notation ("oompa- function?" goes up to Ackermann ordinal = phi(1,0,0,0).

And list of comparisons ordinals with legions and beyond:

phi(1,0,0) ~ {n,n / 2} = n & n & n ... n & n & n phi(1,0,0)+1 ~ {n,n,2 / 2} phi(1,0,0)+2 ~ {n,n,3 / 2} phi(1,0,0)+w ~ {n,n,n / 2} phi(1,0,0)+w^2 ~ {n,n,n,n / 2} phi(1,0,0)+w^w ~ {n,n (1) 2 / 2} phi(1,0,0)+e_0 ~ {n^^n & n / 2} phi(1,0,0)+z_0 ~ {n^^^n & n / 2} 2*phi(1,0,0) ~ {n,n / 3} 3*phi(1,0,0) ~ {n,n / 4} w*phi(1,0,0) ~ {n,n / n} w^2*phi(1,0,0) ~ {n,n / n,n} w^3*phi(1,0,0) ~ {n,n,n / n,n,n} w^w*phi(1,0,0) ~ {n,n / 1 (1) 2} w^(w^w)*phi(1,0,0) ~ {n,n / 1 (0,1) 2} e_0*phi(1,0,0) ~ {n,n / n^^n & n} z_0*phi(1,0,0) ~ {n,n / n^^^n & n} phi(1,0,0)^2 ~ {n,n / 1 / 2} = {n,2 (/1) 2} phi(1,0,0)^3 ~ {n,n / 1 / 1 / 2} = {n,3 (/1) 2} phi(1,0,0)^w ~ {n,n (/1) 2} phi(1,0,0)^2w ~ {n,n (/1) 1 (/1) 2} phi(1,0,0)^(w^2) ~ {n,n (/2) 2} phi(1,0,0)^(w^3) ~ {n,n (/3) 2} phi(1,0,0)^(w^w) ~ {n,n (/0,1) 2} phi(1,0,0)^(w^(w^w)) ~ {n,n (/(1) 1) 2} phi(1,0,0)^e_0 ~ n^^n && n phi(1,0,0)^z_0 ~ n^^^n && n phi(1,0,0)^^2 ~ n && n && n = {n,3 // 2} phi(1,0,0)^^3 ~ n && n && n && n = {n,4 // 2} phi(1,0,1) ~ {n,n // 2} = n && n && n ... n && n && n phi(1,0,2) ~ {n,n /// 2} = n &&& n &&& n ... n &&& n &&& n phi(1,0,w) ~ {n,n (1)/ 2} phi(1,0,w+1) ~ {n,n /(1)/ 2} phi(1,0,2w) ~ {n,n (1)// 2} phi(1,0,w^2) ~ {n,n (2)/ 2} phi(1,0,w^3) ~ {n,n (3)/ 2} phi(1,0,w^w) ~ {n,n (0,1)/ 2} phi(1,0,e_0) ~ {L,X^^X}n,n phi(1,0,z_0) ~ {L,X^^^X}n,n phi(1,0,phi(1,0,0)) ~ {L,L}n,n phi(1,0,phi(1,0,1)) ~ {L,{L,2}}n,n phi(1,0,phi(1,0,phi(1,0,0))) ~ {L,3,2}n,n phi(1,1,0) ~ {L,X,2}n,n phi(1,1,phi(1,0,0)) ~ {L,L,2}n,n phi(1,2,phi(1,0,0)) ~ {L,L,3}n,n phi(1,w,phi(1,0,0)) ~ {L,L,X}n,n phi(1,w+1,0) ~ {L,X,1,2}n,n phi(1,w+2,0) ~ {L,X,2,2}n,n phi(1,w+3,0) ~ {L,X,3,2}n,n phi(1,2w,0) ~ {L,X,X,2}n,n phi(1,3w,0) ~ {L,X,X,3}n,n phi(1,w^2,0) ~ {L,X,X,X}n,n phi(1,w^3,0) ~ {L,X,X,X,X}n,n phi(1,w^w,0) ~ {L,X (1) 2}n,n

Then I noticed the similarity of phi(w^w,0) ~ {X,X (1) 2} & n and phi(1,w^w,0) ~ {L,X (1) 2}n,n. I think that phi(1,alpha,beta) represents legiattic arrays, and:

phi(2,0,0) ~ {L2,1}n,n (lugion arrays) phi(3,0,0) ~ {L3,1}n,n (lagion arrays) phi(4,0,0) ~ {L4,1}n,n (ligion arrays) phi(w,0,0) ~ {LX,1}n,n phi(w+1,0,0) ~ {LL,1}n,n phi(w+2,0,0) ~ {LLL,1}n,n phi(2w,0,0) ~ {(1)L,1}n,n (string of L's) phi(3w,0,0) ~ {(1)LL,1}n,n (add one L in the second row) phi(w^2,0,0) ~ {(2)L,1}n,n (square of L's) phi(w^3,0,0) ~ {(3)L,1}n,n (cube of L's) phi(w^w,0,0) ~ {(0,1)L,1}n,n phi(phi(1,0,0),0,0) ~ {L & L,1}n,n (legion array of L's) phi(phi(2,0,0),0,0) ~ {L2 & L,1}n,n (lugion array of L's) phi(phi(w,0,0),0,0) ~ {LX & L,1}n,n phi(phi(phi(1,0,0),0,0),0,0) ~ {L & L & L,1}n,n phi(1,0,0,0) ~ {L & L & L ... L & L & L,1}n,n (n L's)

And so Bowers' notations not even close to Veblen ordinals.