Jonathan Bowers' multidimensional arrays Sbiis Saibian designed a site with large numbers (first hit with the search Sbiis Saibian)
In section $4.1$ he describes Bowers' notation, but unfortunately he did not come
to the multidimensional arrays yet. In section $3.2.8$ he speaks of an infinity-dimension
space forming a block which represents a well-defined (finite) number.

*

*How can a FINITE number arise from an INFINITE-dimensional space ?
Is this a kind of diagonalization, as it also appears in the fast growing hierarachy ?
 A: For your first question, it is important to note that although the notation uses an infinite dimensional space, for a valid notation only a finite number of entries in the space will be greater than 1. (Bowers uses 1 as the default number rather than the more common 0.)  It's similar to how Bowers' linear arrays can be thought of as having an infinite array of variables as input, but for each notation, you need to only consider a finite number of variables.  However there is no limit to how large that finite number can be.
The infinite dimensional space of natural numbers is equivalent to the ordinal $\omega^{\omega^\omega}$, which is of course well-ordered, so the evaluation rules will eventually terminate, leading to a finite number.
For your second question, yes it is diagonalization very much like in the fast growing hierarchy.  The only significant difference is that at "limit" stages the FGH merely diagonlizes, whereas Bowers adds an extra application of function iteration.  (For example, {a,b,1,d} = {a,a,{a,b-1,1,d},d-1} rather than {a,b,1,d} = {a,a,b,d-1} which would be closer to FGH.)
