That's certainly not the standard. If anything, exponentiation is usually performed right to left, with some exceptions [0]. However, the fact that we even have carets in this conversation isn't because the GP wanted to adopt a left-to-right convention, but due to limitations in the richness of HN's text editor.
If you write a tower of 2^2^9 on a whiteboard and ask 1000 mathematicians and computer scientists to evaluate it, I'm sure 999 or 1000 of them would evaluate it as 2^(2^9).
I'm a mathematician and a computer scientist, so I must be one in a thousand. The link even indicates that ambiguity exists in wild, and I think clear notation would help.
In this case, the absurdity of the number suggests a more realistic number.
Don't you think it's more likely that the original poster made a mistake and that a graph algorithm isn't actually O(2^2^n)? I can't name a single meaningful algorithm that has that time complexity.
