Which is the larger number: one googolplex or Graham's number?
HAHAHAHAHAHAHAHA!!!
Not even close. Not even remotely close. Not even funny, how not even close these two numbers are. They aren’t in the same ballpark, the same world, the same universe, the same multi-universe.
Googolplexian is “ten to the power googolplex”, while googolplex is “ten to the power googol”, and googol is “ten to the power one hundred”. In other words, googol is 1 followed by a hundred zeroes, googolplex is 1 followed by a googol zeroes, and googolplexian is 1 followed by a googolplex of zeroes.
Large?
That’s nothing. Less than microscopic. Negligible. Peanuts, smashed to bits with a sledgehammer and then crushed in a particle accelerator. A single quark, compared with a googolplexian universes, is still far from representing just how tiny googolplexian is compared to Graham’s number.
Seriously, I’m not exaggerating.
Look:
\displaystyle \text{googol} = 10^{100} < \left(3^3\right)^{100} = 3^{300} < 3^{3^{3^3}} = 3\uparrow\!\uparrow 4
The mighty googol is less than 3\uparrow\!\uparrow 4. Remember that a\uparrow\!\uparrow b is just a^{a^{\cdots^a}}, a tower of exponents of height b.
What about googolplex? Well, it’s 10^\text{googol}, and so
\displaystyle \text{googolplex} < \left(3^3\right)^{3\uparrow\!\uparrow 4} = 3^{3 \times 3\uparrow\!\uparrow 4}
Think what you will about 3 \times 3 \uparrow\!\uparrow 4, it’s definitely way, way, waaaaaaaay smaller than 3 \uparrow\!\uparrow 5. Therefore, the mighty googolplex is no more than 3 \uparrow\!\uparrow 6, and for the same reason, the super mighty googolplexian is microscopically smaller than 3 \uparrow\!\uparrow 8.
\displaystyle \text{googolplexian} <\!<\!< 3\uparrow\!\uparrow 8.
Do you know what 3\uparrow\!\uparrow\!\uparrow 3 is? It’s 3\uparrow\!\uparrow(3\uparrow\!\uparrow 3). That’s
3\uparrow\!\uparrow \left(3^{27}\right)
instead of a tower of exponents of height 8, this is a tower of exponents of height 3^{27}. That’s a tower with more than seven trillion 3's. Up there, at its very top, there’s a tiny smidgen with eight 3's; that smidgen alone is more than googolplexian, and we are taking 3 to that power, and then 3 to this, and then 3 to the that again, trillions of times. By the time we’re done, do you still think we can even see googolplexian if we look down with a telescope?
But wait! That’s merely 3\uparrow\!\uparrow\!\uparrow 3. Do you know what 3\uparrow\!\uparrow\!\uparrow\!\uparrow 3 is? That’s 3\uparrow\!\uparrow\!\uparrow (3\uparrow\!\uparrow\!\uparrow 3). It’s an insane recursive construction that is repeated as many times as this gigantic number we’ve just described. There’s no way to even describe in words just how many layers are in this tower of exponents.
But wait! That’s merely the first step in the construction of Graham’s number. The next step involves 3\uparrow\cdots\uparrow 3 with as many arrows as that first step. That’s ridiculously past anything we’ve seen so far. And then Graham’s number does this again, and again and again… 64 times.
As far as “large numbers” go, the construction of googol, googolplex, googolplexian and so on is pathetically, ridiculously small. Even if you continue raising 10 to the previous number a million times, you’re still not even a tiny bit close to 3\uparrow\!\uparrow\!\uparrow 3, and adding a single arrow shoots you up to a completely different level of existence. Please recognize that constructions with Knuth’s up-arrow notation immediately leave far behind any tower of exponents you can still describe.
And of course, there are fast-growing functions that defeat even the up-arrow notation. Large numbers that are sufficiently large to become interesting are really, really, really large.