What does it mean for a number to be independent of ZFC? 
Since the definition of the Busy Beaver function by Radó in 1962, an interesting open question has been what [is] the smallest value of $n$ for which $BB(n)$ is independent of ZFC set theory.

Source: the first sentence of the abstract of the paper A Relatively Small Turing Machine Whose Behavior Is Independent
of Set Theory, with the part in bold added by me (I think it was a typo, they likely forgot the "is").
They proceed to prove that such $n$ is at most 7918.
But $BB(7918)$ is a number, right? So what does it mean for a number to be independent of ZFC?

Bonus question: how much is $BB(7918)$?

Sure, $BB(7918)$ is some number. But it is provably beyond the capabilities of ZFC to figure out which number that is, or even an upper bound for that number.
Specifically, given any (very large, but constructively described) integer $D$, the statement $BB(7918)<D$ can never be proven with ZFC.
As to the bonus question, I've been fascinated with large number notation for a long time. Up-arrow notation, side-arrow notation, Graham's number and the Ackerman function are all cool, and combining them gives some truly mind-bogglingly large numbers. And, of course, one could always invent new, more powerful notation.
Even using recursive functions like the arrow notations mentioned above and the Ackerman function, and constructing new notation like those, I personally believe that there isn't enough room in the observable universe to describe a number anywhere close to $BB(7918)$. And even if it were possible, it's not like we could prove it.
