3^3^3^3^3^3^3^3^3^3^…^3^3^3^3^3^3^3^3^3^…^3^3^3^3^3^3^3^3^3^…^3^3^3^…^3^3^3^3^3^3^3^3…

Credits to Numberphile for the information.

3. It’s a really basic number, isn’t it?

Right? WRONG!

Kevin, Vsauce2

There’s many things you can do with 3. For example, add itself together three times: 3 + 3 + 3 = 9. That’s 3 * 3. But 3 * 3 * 3 is 27, and that’s 3^3. We could say this as 3↑3. That’s 27. 3↑↑3 is just 3↑(3↑3), or 3^3^3. That’s about 7.6 TRILLION. Now, the next step. 3↑↑↑3. That’s 3^3^3^3^3^3^3^3^3^3^3^…^3^3^3^3^… where the stack of 3 is 3↑↑3 long, or 7.6 trillion.

Next step is 3↑↑↑↑3, and that’s just 3^3^3^3^3^3^3^3^3^3^3^3^3^3^3^…^3^3^3^3^3^3^3^3^3^…^3^3^… where the stack of 3 is 3↑↑↑3 long, or… whatever it is.

The fun’s just beginning.

Let g₁ be 3↑↑↑↑3. g₂ would be 3↑↑↑↑↑↑↑↑↑↑↑…↑↑↑↑↑↑↑↑…↑↑↑↑3, where the number of ↑ is g₁. g_n is 3↑↑↑↑↑↑↑↑↑↑↑↑…↑↑↑↑↑↑↑↑↑…↑↑↑↑…↑3 where the number of ↑ is g_n-1.

What is Graham’s number?

It’s g₆₄.

According to Numberphile, if you tried to write down Graham’s number or visualise it, your head would apparently not have enough capacity to hold that much information, and your brain would collapse into a black hole. Weird, isn’t it?

But there’s more.

TREE(3).

Also, thanks to Numberphile again for the information.

Actually, never mind. That’s a post for another time.