Saibian demonstrates that for very large numbers $N$, $N\uparrow\uparrow N$ is only "slightly larger" than $N$.
I would like to demonstrate it for the number
$$N=4\uparrow^4 4=4\uparrow^3 4\uparrow^3 4\uparrow^3 4$$
I want to bound $N\uparrow\uparrow N$ from above. I think $4\uparrow^5 4$ would be an upper bound, but even if this is the case, I would like to find a better upper bound.
For which $k$ do we have $4\uparrow\uparrow\uparrow k>N\uparrow\uparrow N\ $ ?
The value $k$ should be near $4\uparrow^3 4\uparrow^3 4$. This would show that $N\uparrow\uparrow N$ is "not much larger" than $N$.