User blog comment:Edwin Shade/How Do I Evaluate BEAF Arrays In Two Dimensions ?/@comment-30754445-20170827183955/@comment-5529393-20170829045217

Note that even the single variable array {n} can "get arbitrarily high", since we can make n as big as we choose. So no notation or function can beat {n} in the sense that its values can be larger than {n} for any value of n, since {n} goes to infinity. But, you can make functions that are faster growing than {n}; for example, {n,n} is a faster growing function than {n}. Then {n,n,n} is faster growing than {n,n}, and {n,n,n,n} is faster growing than {n,n,n}, and so on up to infinity. At this point, one might ask a similar question to the one you asked: Since we can make {n,n,n,n,...,n} for any number of n's, doesn't that go "all the way up"? The answer is no, in the sense that we can make a function that is faster growing than all of the {n,n,n,n,...,n}'s. If we define f_m(n) = {n,n,n,...,n} with m n's, then the function f(n) = {n,n,n,...,n} with n n's will be faster growing than all the f_m's, since f will catch up to f_m at m, and be ahead from m+1 on. Note that f(n) is just {n,n (1) 2} in Bowers' notation.

Similarly, if we define g_m(n) as {n,n (m) 2} than we can beat all the g_m's by defining g(n) = {n,n (n) 2}. This is equal to {n,n (1,2) 2} in Bowers' notation. (NB: Actually, Bowers only defines his notation precisely up to dimensional arrays; beyond that, his definition is given vaguely in terms of "structures", but it's not clear what those structures are, much less how we are supposed to write them down. However, there seems to be a reasonably natural way to extend his notation up to "tetrational arrays", so I think the general consensus here is that BEAF is well-defined up to that point.  I haven't taken a poll or anything though.)  Then, you can keep going to {n,n,2 (1,2) 2} and so on.

Some terminology: We have taken to calling the parts of an expression bounded by parentheses "separators". Also "," is a separator, the lowest separator, and can be considered shorthand for (0). Dimensional arrays are those arrays where all the separators are just natural numbers, like {a,b,c (1) d,e (3) f,g,h (4)(2) i}. Beyond separators of the form (a), we have linear array separators like (a,b,c,d,e), and then separators with higher dimensions like (2 (2) 4 (3) 8 (5)(3) 2 (1) 2); i.e. the separators are themselves dimensional arrays. Arrays where the separators are dimensional arrays are called superdimensional arrays by Bowers; these are the arrays where you can have separators within separators, i.e. the nesting of separators can go 2 deep. Then trimensional arrays are arrays where the separators are superdimensional arrays, i.e. the nesting of separators can go 3 deep. Then you have quadradimensional arrays for nesting 4 deep, and so on. Arrays that can have any level of nesting are called tetrational arrays by Bowers.