Definite upvote. That essay is insanely fascinating; I tend to re-read it every few months.

I like the article, too, but I'm always disappointed when it doesn't cover Knuth's up-arrow, which seems absurdly relevant.

Thanks. Obviously http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

Shorter read: http://www.daviddarling.info/encyclopedia/K/Knuths_up-arrow_... , which comes quickly to next step:

Three up-arrows together represent a still more vastly powerful operator, equivalent to hyper5 or pentation, or a power tower of power towers

So if humans can compute the first few Busy Beaver numbers, does that mean we're not Turing machines?

Not at all. Just because a problem in general is uncomputable does not mean that certain specific instances can't be computed. For example, the halting problem is undecidable, but anyone can see that loop of the form

    while (1) { }

is an infinite loop. Undecidability and uncomputability apply only at the level of general problems, not specific instances.

Thanks :)