Inspired by: sites.google.com/view/thaawesomes-big-numbers/home (Humongous Numbers)
Disclaimer: This notation is ill-defined as a celebration for April Fool's Day.
All variables must be non-negative integers.
a[[0]]b = a[1]b = a^b
a[c]b = a^^^^^...^^^^^b with c arrows
a[[1]]b = a[b]a
a[[1] + 1](b + 1) = a[[1]](a[[1] + 1]b)
a[[1] + 2](b + 1) = a[[1] + 1](a[[1] + 2]b)
a[[1] + 3](b + 1) = a[[1] + 2](a[[1] + 3]b)
a[[1] + n + 1](b + 1) = a[[1] + n](a[[1] + n + 1]b)
a[[1][1]]b = a[[1] + b]a
a[[1][1] + 1](b + 1) = a[[1][1]](a[[1][1] + 1]b)
a[[1][1][1]]b = a[[1][1] + b]a
a[[1][1][1][1]]b = a[[1][1][1] + b]a
a[[1]*c]b = a[[1][1]...[1][1]]b with c copies of [1]
a[[2]]b = a[[1]*b]a
a[[2] + 1](b + 1) = a[[2]](a[[2] + 1]b)
a[[2][1]]b = a[[2] + b]a
a[[2][2]]b = a[[2][1]*b]a
a[[3]]b = a[[2]*b]a
a[[4]]b = a[[3]*b]a
a[[c + 1]]b = a[[c]*b]a
a[[[1]]]b = a[[b]]a
a[[[[1]]]]b = a[[[b]]]a
a[0, 1]b = a[[[...[[[a]]]...]]]a with b nests of square brackets
a[1, 1](b + 1) = a[0, 1](a[1, 1]b)
a[[1], 1]b = a[b, 1]a
a[[0, 1], 1]b = a[[[...[[a]]...]], 1]a with b nests of square brackets in [x, 1]
a[0, 2]b = a[[[...[[[0, 1], 1], 1]..., 1], 1], 1] with b nests of square brackets
a[0, 3]b = a[[[...[[[0, 2], 2], 2]..., 2], 2], 2] with b nests of square brackets
a[0, [1]]b = a[0, b]a
a[0, [0, 1]]b = a[0, [[...[[a]]...]]]a with b nests of square brackets in [0, x]
a[0, 0, 1]b = a[0, [0, [0, ...[0, [0, [0, 1]]]...]]] with b nests of square brackets
a[0, 0, 0, 1]b = a[0, 0, [0, 0, [0, 0, ...[0, 0, [0, 0, [0, 0, 1]]]...]]] with b nests of square brackets
a[0 / 1]b = a[0, 0, 0, ..., 0, 0, 0, 1]a with b entries of 0
a[0, 1 / 1]b = a[[...[[0 / 1] / 1]... / 1] / 1] with b nests of square brackets in [x / 1]
a[0 / 2]b = a[0, 0, 0, ..., 0, 0, 0, 1 / 1]a with b entries of 0 before /
a[0 / n + 1]b = a[0, 0, 0, ..., 0, 0, 0, 1 / n]a with b entries of 0 before /
a[0 / 0, 1]b = a[0 / [0 / ...[0 / [0 / 1]]...]]a with b nests of square brackets
a[0 / 0 / 1]b = a[0 / 0, 0, 0, ..., 0, 0, 0, 1]a with b entries of 0 after /
{0} is ",", and {1} is "/"
a[0 {2} 1]b = a[0 / 0 / ... / 0 / 0 / 1]a with b entries of 0
a[0 {3} 1]b = a[0 {2} 0 {2} ... {2} 0 {2} 0 {2} 1]a with b entries of 0
a[0 {n + 1} 1]b = a[0 {n} 0 {n} ... {n} 0 {n} 0 {n} 1]a with b entries of 0
a[0 {0, 1} 1]b = a[0 {[0 {...[0 {[0 / 1]} 1]...} 1]} 1]a with b nests of square brackets
Now let's define another function.
f(0) = a[1]b
f(1) = a[0, 1]b
f(2) = a[0, 0, 1]b
f(3) = a[0 {0, 1} 1]b
f(4) = a[0 {0, 0, 1} 1]b
f(5) = a[0 {0 {0, 1} 1}]b
f(6) = a[0 {0 {0, 0, 1} 1} 1]b
f(n) = a[0 {0 ... {0 {0, 1} 1} ... 1} 1]b from the above sequence
a[0 \ 1]b = f(b)
a[0 \ 2]b = a[f(b) \ 1]a
a[0 \ 0 \ 1]b = a[0 \ f(b)]a
a[0 | b] = a[0 \ 0 \ ... \ 0 \ f(b)]a with f(b) \'s
a[0 - b] - a[0 | 0 | ... | 0 | f(b)]a with f(b) |'s
Now, let's define yet another function.
f(0, 1) = a[0 \ 1]b
f(0, 2) = a[0 | 1]b
f(0, 3) = a[0 - 1]b
a[0`]b = f(a, b)
f(0, 0, 1) = a[0`]b
f(0, 1, 1) = a[0 \ 1`]b
f(0, 2, 1) = a[0 | 1`]b
f(0, 3, 1) = a[0 - 1`]b
a[0``]b = f(0, b, 1)
a[0```]b = f(0, b, 2)
f(0, 0, n) = a[0```...```]b with n `'s
f(0, 0, 0, 1) = f(0, 0, f(0, 0, ...f(0, 0, f(0, 0, 1))...)) with a f's
f(0, 0, 0, 0, 1) = f(0, 0, 0, f(0, 0, 0, ...f(0, 0, 0, f(0, 0, 0, 1))...)) with a f's
f(0 / 1) = f(0, 0, 0, ..., 0, 0, 0, 1) with a 0's
Let's define the g function!
g(1) = f(0, 1)
g(2) = f(0, 0, 1)
g(3) = f(0 {0, 1} 1)
g(4) = f(0 {0, 0, 1} 1)
g(0, 1) = g(g(...g(g(1))...)) with a g's
And then the h function.
h(1) = g(0, 1)
h(2) = g(0, 0, 1)
h(3) = g(0 {0, 1} 1)
h(4) = g(0 {0, 0, 1} 1)
And now...
j(1) = f(1)
j(2) = g(1)
j(3) = h(1)
j(0, 1) = j(j(...j(j(1))...)) with a j's
k(1) = j(0, 1)
k(2) = j(0, 0, 1)
k(3) = j(0 {0, 1} 1)
l(1) = k(0, 1)
l(2) = k(0, 0, 1)
m(1) = j(1)
m(2) = k(1)
m(3) = l(1)
n(1) = j(1)
n(2) = m(1)
p(1) = j(1)
p(2) = n(2)
q(1) = j(1)
q(2) = p(2)
And even crazier...
r(1) = j(2)
r(2) = k(2)
r(3) = l(2)
r(4) = m(2)
r(5) = n(3)
r(6) = p(3)
r(7) = q(3)
s(1) = r(r(...(r(r(1)))...)) with b r's
t(1) = s(s(...(s(s(1)))...)) with b s's
a[u]b = s(s(...(s(s(1)))...)) with b t's
a[2u](b + 1) = a[u](a[2u]b)
a[u(1,1)]b = a[u(x)]a, where x is a[u]b
a[u(2,1)]b = a[u(x)]a, where x is a[u(1,1)]b
a[u(1,2)]b = a[u(x,1)]a, where x is a[u(1,1)]b
a[u(1,3)]b = a[u(x,2)]a, where x is a[u(1,2)]b
a[u(1,1,1)]b = a[u(1,x)]a, where x is a[u(1,b)]a
a[u(1,1,1,1)]b = a[u(1,1,x)]a, where x is a[u(1,1,b)]a
a[u(1/1)]b = a[u(1,1,1,...,1,1,1)]a, with b 1's
a[u(1/2)]b = a[u(1,1,1,...,1,1,1/1)]a, with b 1's before /
a[u(1/1/1)]b = a[u(1/1,1,1,...,1,1,1)]a, with b 1's after /
We can even do a[u(1/1/1/.../1/1/1)]b, and that, is the limit of the Madom's Awesomer notation. That limit is approximately the Takeuti-Feferman-Buchholz ordinal (TFBO) level.