[shhsshs]

`9↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑9` seems like a reasonable guess (barring encoding cheats/trickery like @masfuerte commented!)

Edit: I've misread the above comment and my number is is 64 bytes (significantly more than 64 bits. The largest 64 bit number through my approach would be `9↑↑↑↑↑↑9`, which is significantly smaller.

[lisper]

I can do you one better and specify that the normal base-2 integer represented by the bits is the number of up-arrows. But as /u/tromp has already pointed out, that is not very interesting.

[tuhgdetzhh]

Is there any intuition on how big this number is?

[tromp]

In terms of the Fast Growing Hierarchy, it's about f_62(9) or what the article would denote as [62] 9. It's way smaller than Graham's Number, which involves 64 iterations of mapping n to 3 ↑↑↑... {n uparrows) 3, whereas this expression has between 1 and 2 iterations.

[odo1242]

If every atom in the universe had a universe inside each proton, there still wouldn’t be enough atoms within all the universes in the protons. In fact you might not make it to four arrows with the above lol