This is my very first part of my notation.
The basic array notation has the following form:
a[c]b
where all variables (a, b, c) are non-negative integers.
Base Rule: a[0]b = a×b
Prime Rule: a[c]1 = a, a[c]0 = 1 if c > 0, a[0]0 = 0
Recursion Rule: a[c]b = a[c-1](a[c](b-1)) if b > 1 and c > 0
If there are two or more distinct rules to apply to a single expression, the lowest-numbered rule which is applicable and whose result is a valid expression will be applied.
It's easy to see that the above three rules mirror the definition of up-arrow notation. So the limit of this notation is f_ω(n).
A small example:
2[3]3
= 2[2]2[3]2
= 2[2]2[2]2[3]1
= 2[2]2[2]2
= 2[2]2[1]2[2]1
= 2[2]2[1]2
= 2[2]2[0]2[1]1
= 2[2]2[0]2
= 2[2]2*2
= 2[2]4
= 2[1]2[2]3
= 2[1]2[1]2[2]2
= 2[1]2[1]2[1]2[2]1
= 2[1]2[1]2[1]2
= 2[1]2[1]2[0]2[1]1
= 2[1]2[1]2[0]2
= 2[1]2[1]2*2
= 2[1]2[1]4
= 2[1]2[0]2[1]3
= 2[1]2[0]2[0]2[1]2
= 2[1]2[0]2[0]2[0]2[1]1
= 2[1]2[0]2[0]2[0]2
= 2[1]2[0]2[0]2*2
= 2[1]2[0]2[0]4
= 2[1]2[0]2*4
= 2[1]2[0]8
= 2[1]2*8
= 2[1]16
= 2[0]2[1]15
= 2[0]2[0]2[1]14
= ...
= 2[0]2[0]2[0]...[0]2[0]2[0]2 (with 16 2's)
= 2^16
= 65,536