I invented another new notation that generalizes Steinhaus-Moser notation and achieves BEAF-level growth rates. The first delimiter, a(*^b)c, is defined like so:

a(*^2)b = a^b

a(*^3)b = \(a^{a^b}\)

a(*^n)b = \(a^{a^{...^{a^b}}}\) w/ n-1 copies of a

a(*^*)b = a(*^b)b

a(*^^2)b = a^a repeated b times (equal to a in b triangles in Steinhaus-Moser notation). a(*^^3)b = \(a \uparrow\uparrow 3\) repeated b times (equal to a in b triangles in a version of Steinhaus-Moser notation where n[3] = n^(n^n) instead of just n^n) a(*^^n)b = \(a \uparrow\uparrow n\) repeated b times a(*^^*)b = \(a \uparrow\uparrow a\) repeated b times

a(*^^^2)b = a(*^^2)a repeated b times (this is equivalent to a in b squares in Steinhaus-Moser notation)

Mega can be expressed exactly as 2(*^^^2)2 or 2(*^^^^2)1. In fact, a in a b-sided polygon in Steinhaus-Moser notation can be expressed as a(*^^^...^^^2)a w/ b-2 stars.

I defined a series of numbers analogous to the googol, starting with bidugol, which is equal to 10(*^^2)100 (approximately E100,000,000,011#99). I then named the tridugol, which is equal to 10(*^^3)100 (approximately E10,000,000,010#199), the quadugol (10(*^^4)100), and the quindugol, which is equal to 10(*^^5)100.

The hypermega is equal to 2(*^^^3)2. I actually came up with the name back in July 2015 (long before I created my site, let alone joined this wiki or even conceived of this notation). This number is approximately \(10 \uparrow\uparrow (2^{2^{66} + 1} + 3)\), or more precisely, about \((10 \uparrow)^{2^{2^{66} + 1}} 2.9374371566719126 \times 10^{22212093154093428548}\), and its last 18 digits are ...,176,575,436,964,233,216. Similarly, the quadramega is equal to 2(*^^^4)2, the quintimega is equal to 2(*^^^5)2, and the star mega is equal to 2(*^^^*)2.

Next, I defined a(*^[0]^b)c to be equal to a(*^^^...^^^b)a w/ c ^s, which diagonalizes over everything I covered up to this point. After this, the notation gets a bit complicated, so I linked my site's article for more. If my analysis is correct, the highest delimiter I have currently defined, *^[*, *, 1]^*, has a growth rate roughly equivalent to \(f_{\omega^{\omega^{\omega^2}}}(n)\) in the fast-growing hierarchy, on par with Bowers' 2-superdimensional arrays.

Source:

Target growth rates:

a(*^[*, *, *]^b)c: \(\omega^{\omega^{\omega^\omega}}\)

a(*^[[0]]^b)c: \(\varepsilon_0\)

a(*^[[1]]^b)c: \(\zeta_0\)

a(*^[[2]]^b)c: \(\eta_0\)

a(*^[[*]]^b)c: \(\varphi(\omega, 0)\)

a(*^[[*, 1]]^b)c: \(\varphi(\omega^\omega, 0)\)

a(*^[[*, *]]^b)c: \(\varphi(\varphi(1, 0), 0)\) or \(\varphi(\varepsilon_0, 0)\)

a(*^[[*, *, *]]^b)c: \(\varphi(\varphi(\varepsilon_0, 0), 0)\)

a(*^[[[0]]]^b)c: \(\Gamma_0\)

a(*^[[0]]^b)c: \(\varphi(2, 0, 0)\)

Diagonalization over the number of pairs of square brackets: \(\varphi(\omega, 0, 0)\)