Numbers that are expressed using Knuth's up-arrow notation can be difficult to compare. Say you have a number N such that 10↑↑15 < N < 10↑↑16. 10↑↑15 and 10↑↑16 look like they should be pretty close, because 15 and 16 are pretty close. But really 10↑↑16 is way way bigger than 10↑↑15, so the given bounds on N aren't very helpful.
This article proposes a notation to improve the granularity of up-arrow notation.