Ascending Notation is a notation by me involving numbers increasing.
First, n+ is all the naturals from 1 to n concatenated, including n.
For example, 5+ = 12345, 11+ = 1234567891011.
Next, we define n++ as (n+)+, or in other words, the naturals from 1 to n+ concatenated.
2++ = 12+ = 123456789101112
3++ = 123+
4++ = 1234+
We can define the triple-plus similarly.
2+++ = 12++ = 123456789101112++ which is HUGE for just 2.
And we need to diagonalize. Let's define n[2] as n++...++ with n pluses.
2[2] = 2++ = 123456789101112
3[2] = 3+++ = HUGE
n[3] = ((n[2])[2]...)[2] with n [2]s
2[3] = 2[2][2] = 123456789101112++...++ with 123456789101112 pluses
Define n[4] similarly to n[3]
Next we have n[1,2] = n[n]
n[2,2] = n[1,2][1,2]...
n[1,3] = n[n,2]
n[1,4] = n[n,3]
n[1,1,2] = n[n,n]
n[1,1,1,2] = n[n,n,n]
limit is n[n,n,n...n,n,n]. I could extend this, but i don't feel like it.
It's also very hard to analyse, as it uses concatenation which has no set growth rate.
FORMAL DEFINITION
# is the unchanged remainder of the expression,
[#]^n is [#] strung together n times
n+ = all numbers 1 to n concatenated together.
n[1] = n+
n[#,1] = n[#]
n[m+1,#] = n([m,#]^n)
n[1,1,1...1,1,m+1,#] = n[n,n,n...n,n,m,#]
IMPORTANT NOTE: n[#1][#2]...[#k-1][#k] is interpreted as (((n[#1])[#2])...[#k-1])[#k]