This past Wednesday's What-If had this image at the bottom:
In particular, I am interested in $20 \uparrow\uparrow\uparrow\uparrow 20$. I immediately thought of Graham's Number, but clearly that one is much, much larger. So then I thought of $G_1 = 4 \uparrow\uparrow\uparrow\uparrow 4$, but that one is clearly much smaller.
However, I'm not sure whether $4 \uparrow\uparrow\uparrow\uparrow\uparrow 4$ is larger or smaller than $20 \uparrow\uparrow\uparrow\uparrow 20$. On the one hand, the former has an extra level of iteration, but on the other hand, the $20$ in the latter could perhaps result in a rate of growth large enough to surpass the former. So, which is greater, and how can we know?
$4\uparrow\uparrow\uparrow\uparrow\uparrow4$ is far far larger than $20\uparrow\uparrow\uparrow\uparrow20$.
For convenience, I will use $\uparrow^n$ for $\underset{n}{\underbrace{\uparrow\uparrow\cdots\uparrow}}$. We want to know the following: $$4\uparrow^{5}4=\boldsymbol4\uparrow^{4}\left(4\uparrow^{4}\left(4\uparrow^{4}4\right)\right)\overset{?}{>}\boldsymbol{20}\uparrow^420$$ So the question is, is $4\uparrow^{4}\left(4\uparrow^{4}4\right)$ bigger than $20$ by enough to offset the difference between the "bases"? To figure out the answer to this question, we can look at operations that grow more slowly than $\uparrow^4$ and look for a pattern. Let's start with multiplication: what value of $x$ makes $4*x>20*20$? $x=101$ would work.
What value of $x$ makes $4^x>20^{20}$? Well, these numbers are small enough for a good calculator to tell us, but if we just want a ballpark estimate, note that $\log_420<2.5$, so $20^{20}<(4^{2.5})^{4^{2.5}}=(4^{2.5})^{32}=4^{80}$. Even our terrible estimate of $80$ shows that the added iteration in exponentiation decreased the $x$ that we need from $101$ (in fact, $4^{44}>20^{20}$).
As we go into deeper levels of iteration, any value of $x$ that's at least $80$ should work, for sure. So to answer the question, all we really need to know is whether or not $4\uparrow^{4}\left(4\uparrow^{4}4\right)\ge80$. But even for the far smaller number $4^4$, we have $4^4=256>80$.
With a calculator like Wolfram|Alpha, we can afford to use more brute force. Let's just look at a lower-iteration version of the question: $4\uparrow^24$ vs. $20\uparrow20=20^{20}$. $$20^{20}=104\,857\,600\,000\,000\,000\,000\,000\,000$$ But $4\uparrow^{2}4\gg4\uparrow^{2}3=4^{\left(4^{4}\right)}=4^{256}$ $$=13\,407\,807\,929\,942\,597\,099\,574\,024\,998\,205\,846\,127\,479\,365\,820\,592\,393\,377\,723\,561\,443\,721\,764\,030\,073\,546\,976\,801\,874\,298\,166\,903\,427\,690\,031\,858\,186\,486\,050\,853\,753\,882\,811\,946\,569\,946\,433\,649\,006\,084\,096$$