No only that, but the number of digits in G_64 (~log10(G_64)) exceeds the number of Planck volumes (V_p) in our universe too. As does the number of digits in that number (~log10^2(G_64)). And so on and so on, for a number of times itself exceeding the number of Planck volumes in the universe.
P_v ~ 4.65e185
P_v < G_64
P_v < log10(G_64)
P_v < log10^(2)(G_64)
…
P_v < log10^(P_v)(G_64)
G_64 < BB(18)
That said, we can grow bigger still. Scott Aaron defined a closely-related concept called a Beeping Busy Beaver that grows uncomputably even if we had an oracle for BB(n). That is, even if we had access to a magic genie that could tell us any BB(n), this function BBB(n) even still grows uncomputably. It’s almost as if there’s “sizes” of uncomputability much the same as there are distinct sizes of infinity.
We don't know if it exceeds BB(5) either, do we?
In the functional realm [1], we know:
BB_λ(36) < 5 * G_64 + 6 <= BB_λ(114)
Note that these are measured in bits, and that it would take 18 * 2 * (2 + ceil(log(18))) = 252 bits to represent the 18-state TM computing G_64.
The 847 state TM whose halting is known to be unprovable in ZFC takes 847 * 2 * (2 + ceil(log(847)) = 20328 bits to describe. Meanwhile, in a mere 215 bits, we can encode a Laver table program [2] whose halting is not known to be provable in ZFC [3]. I.e. it may or may not be unprovable, we don't know yet.
[2] https://github.com/tromp/AIT/blob/master/laver.lam
[3] https://codegolf.stackexchange.com/questions/79620/laver-tab...
There is, isn't there? A TM plus an oracle can solve the halting problem for the original TM but is just as susceptible to a halting problem of its own, and this goes on ad infinitum.