User blog comment:Simplicityaboveall/The Construction of Extremely Large Numbers/@comment-11227630-20160726150520/@comment-11227630-20160729060432

"$$\varepsilon_1$$ is approx. $$\omega\uparrow\uparrow(\omega2)$$" - that depends on how you define tetration on ordinals.

If ordinal tetration is defined as follows: then for all $$\alpha\ge\omega$$, $$\omega\uparrow\uparrow\alpha=\verepsilon_0$$, because $$\omega^{\varepsilon_0}=\varepsilon_0$$. And we still don't know what's "w^^(w+1)".
 * 1) $$\alpha\uparrow\uparrow0=1$$
 * 2) $$\alpha\uparrow\uparrow(\beta+1)=\alpha^{\alpha\uparrow\uparrow\beta}$$
 * 3) $$\alpha\uparrow\uparrow\beta=sup\{\alpha\uparrow\uparrow\gamma|\gamma<\beta\}$$ iff $$\beta$$ is a limit

Here's an idea for solutions - try down-arrow notation. For ordinals, a+b can be larger or equal to b, but a+b must be larger than a. So a left-associative notation (such as down-arrow notation) fits better.