BusyBeaver(6) Is Quite Large

>imagine you had 10,000,000_10 grains of sand. Then you could … well, uh … you could fill about 10,000,000_10 copies of the observable universe with that sand. I hope that helps people visualize it!

People can't visualize numbers that big. There's more ways to express numbers than just counting them. For example a single grain of sand has infinite states it can be in (there are an infinite amount of real numbers), so you could say a single grain of sand could represent BB(6). Combinations can grow exponentially, so that may be something useful to try and express it.

At some point big numbers become much more about the consistency strength of formal systems than “large quantities”.

I.e., how well can a system fake being inconsistent before that fact it discovered? An inconsistent system faking consistency via BB(3) will be “found out” much quicker than a system faking consistency via BB(6). (What I mean by faking consistency is claiming that all programs that run longer than BB(n) steps for some n never halt.)

I'm confused about this example, isn't the count of grains of sand equal to the count of observable universes so it'd be a single grain of sand per universe?

The "about" does a lot of heavy lifting in this example. Dividing 10,000,000_10 by the number of grains that fit into one universe doesn't change it much. The 10,000,000 would get smaller somewhere in the deep depths of the decimal fraction.

If the universe rounds to the nearest Planck unit, then a grain of sand suddenly has not all that many states.

Using infinite precision to make things seem tractable is sleight of hand in my book. Stick with integers when you're describing scale.