TREE(3) is significantly larger, though the explanations as far as I've found them require understanding much more complicated maths than for Graham's number (at least, I can follow the explanation for Graham's number, but not TREE(3)).

The fun bit about TREE(3) is that the sequence TREE(1), TREE(2), TREE(3) goes :

TREE(1): 1

TREE(2): 3

TREE(3): explosion

n(4) is another fun one. n(3) is less than Graham's number, which itself is roughly A^64(4). n(4) is about A^A^(187196)(1)

http://everything2.com/title/TREE%25283%2529

Am I understanding correctly we don't know how large TREE(3) is?

I've only ever seen lower bound values given.

E.g. http://mathoverflow.net/questions/93828/how-large-is-tree3

If you've taken the time to figure out Graham's number, take a few minutes to familiarise yourself with the Ackerman function, so that you can grasp the full weight of this xkcd: https://xkcd.com/207/

do you mind linking to what you read on Graham's number? The wikipedia is interesting, but it seems like you found something more so

Not OP, but Wait But Why did a great article on it a few weeks ago:

http://waitbutwhy.com/2014/11/1000000-grahams-number.html

I have always enjoyed Numberphile and here is the man himself (Graham) explaining it https://www.youtube.com/watch?v=GuigptwlVHo

If you like big numbers, this article is a must read:

http://www.scottaaronson.com/writings/bignumbers.html

Wildberger recently provided a more systematic approach (though Graham's number is not mentioned explicitly) to hyperoperations in one of his videos [0]. He also discusses some implications of working with numbers so large (including in other videos of the series).

[0] https://youtu.be/Wv65xhrJ0zc

you might enjoy reading the analysis of loader.c number here:

http://djm.cc/bignum-results.txt