Project: $$\omega^{\omega^{8}}$$ in Snap!
I made a new notation notated as SSSS[index] in Snap! Here are script pics of how it works:
In words:
Take a string whose digits range from 1 to 9 and whose first digit is 1, and assign an index to it. The full format should look like SSSS[index]. Now do the following:
If the string you took is "1", return the index+1.
If the string isn't "1", but ends with 1, then return SSSS<string without 1>[SSSS<string without 1>[...]] the index amount of times, with the index at the end of the recursion. For example, SSSS<11>[4] = SSSS<1>[SSSS<1>[SSSS<1>[SSSS<1>[4]]]] and SSSS<121231>[4] = SSSS<12123>[SSSS<12123>[SSSS<12123>[SSSS<12123>[4]]]]
Otherwise, do the following process until it ends with a 1:
If the string ends with "2", and the full string only has one "1", which is at the beginning, then return the full string without the "2" repeated the index amount of times.
If the string ends with "2", but the full string has more than one "1", then find the last "1" in the string, group it with the digits after it, and the apply this method and ungroup the group.
Otherwise, the last digit is duplicated into duplicates of [digit - 1], duplicated the index amount of times.