What Numbers there are

$3$, $4\cdot4$, $3i$, $\sqrt{2}$, $2^{1000}$, and $BB(9)$ are all numerals, symbols for numbers. Symbols are inert and do not intrinsically point at particular numbers (what would it even mean to intrinsically point?). Rather they encode processes, computations, by which to produce or identify numbers.

For instance the numeral $3$ encodes, as mere hardcoded associations since the symbol $3$ does not have informative decomposable structure, any of many structures with well established relationships to the structure resulting from the next step of Peano-like counting beyond that named by $2$, where the choice of specific structure is up to the purposes of the interpreter at that moment, and where the abuse of notation is justified by the properties of the well established relationships.

$4\cdot4$ is a composite symbol that can be decomposed into instructions. Those instructions can be fed into many different computations. We may wish to interpret the data in the context of its inclusion in an algebraic equation and, say, perform some cancellation. Or we may wish to evaluate the fully applied multiplication function (* 4 4) (written lisp-y to disambiguate), which is another symbol manipulation process.

Similarly $BB(9)$ names a structure, specifically it is a symbol used to refer to the expanded standard definition of the ninth busy beaver number. The ninth busy beaver number is exactly the definition which gives computationally useful information. Notably the definition of the ninth busy beaver number does not give a digit string with a claim of correspondence to the definition. Similarly $4\cdot4$ does not come packaged with $16$, it takes additional computational work to determine that a correspondence we care about (one with nice properties) exists between the process named by $4\cdot4$ and that named by $16$.

$4\cdot4$ exists, it's the symbol just to the left of where your eyes are pointing. $16$ exists similarly. Processes mapping back and forth between those symbols exist and have nice properties, this claim would be justified by exhibiting them, which is left as an exercise to the reader.

It is an independent fact that many real world computations (computations where we choose symbols with which to compute based on things outside the symbol manipulation game) produce similar final results. Which is what we mean when we say "I have sixteen sheep": I applied a computation to my perceptions which eventually mapped down to symbols which have relationships with nice properties to the numerals.

Confusion about whether a number exists comes from imprecision in what one means by the number in question and what one means by existence. Be precise about both and the answer is always easy. The symbol $BB(9)$ exists, demonstratum est. The definition that we've agreed to use $BB(9)$ for exists, https://en.wikipedia.org/wiki/Busy_beaver. Does the number $BB(9)$ exist? Of course, the aforelinked definition is the number, but that's not a digit string. I don't have a digit string with a proof of equivalence to that definition on hand, and I'm certain I never will since the digit string required would be too large to fit in my world. Nor do I have an arithmetic (not even if extended with things like Knuth arrows) expression with an equivalence proof, though someone may produce one at some point.

You didn't ask about the digit string directly because you couldn't ask about the digit string directly. But is there such a digit string? Instantiated, no. Might it be possible to produce such a digit string (and accompanying proof)? Not here. Some other more complex expression with an equivalence proof? Maybe, I hope so but no one has done it yet.

Be precise about what the thing you're asking about the existence of is and you'll find that you fail to refer to things that don't exist. You may have imagined that there were many things you knew of that are beyond existence, but once you become precise you will see that the only things about which you were sufficiently precise to refer to were those whose existence is clear.