raising to the same power

Question

• when we add the same number many times we can use multiplication for shorter: $$2+2+2+2 = 2\times4$$
• when we multiply the same number many times we can use exponentiation for shorter: $$2\times2\times2\times2 = 2^4$$
• is there next level of shorter form equivalent to rising to the same power over and over again? $$2^{2^{2^2}} = ?(2,4)$$ In the above example there is not much of shortening because we can mulitply powers and have $$2^8 = ?(2,4)$$ But look at this: $$1000^{1000^{1000^{1000^{1000^{1000}}}}} = 1000^{1000000000000000} = ?(1000,6)$$
• if such a operation exists what about next levels of shortening?

Answer 1

I believe that you're after something like Knuth's up-arrow notation. This has been explained on this site. See How exactly does Knuth's Up-Arrow notation work?