This reminded me of this
XKCD comic (3rd panel).
Graham's number is very very large. Knuth's up arrow notation is used to describe how large this number really is. Multiplication is just repeated addition of one number again and again and again, the same way exponentiation is repeated multiplication again and again. Knuth decided to extend this concept to tetration, which is repeated exponentiation, and so on and so forth:
a+a+a...a+a+a = b*a (multiplication)
\-----v-----/
b many a's
a*a*a...a*a*a = b^a (exponentiation)
\-----v-----/
b many a's
a^a^a...a^a^a = b^^a (tetration)
\-----v-----/
b many a's
a^^a^^a...a^^a^^a = b^^^a (ect.)
\-------v-------/
b many a's
Forgive my horribly ASCII curly brackets. Grahams number is defined as such:
3^^... ...^^3 \
\--------v---------/ |
|
3^^... ...^^3 |
\-------v-------/ |
. |
. > 64 layers.
. |
3^^... ...^^3 |
\-------v------/ |
|
3^^^^3 /
So it is very large indeed. Then putting it into the ackermann function (which increases very fast) is horrifying prospect. Even so, there is another notation, called conway's chained arrow notation which makes numbers increase even faster. Even so, these numbers are computable and real and can be referenced.
This brings me to the topic of some truly terrifying numbers which grow larger than any computable function in existence: Busy beaver numbers. The halting problem in mathematics has been proven to be incomputable. The problem is whether or not a turing machine with n operational states will ever halt. The problem is proven impossible for any computer to figure out.
The busy beaver number is the number of operations it takes for the longest running program on a turing machine before it halts. Therefore, we know that if a turing machine runs longer than this, it will never halt, and run indefinitely.
If we knew what the busy beaver number was for an n state turing machine, we could simply run the program, and if it runs for more operations than that number, we know it will never halt. Thus we would have a foolproof solution to the halting problem. Unfortunately, we already know a solution doesn't exist, so we can't know what these numbers are. Similarly, if we could compute a number larger than the busy beaver number, we could use that number instead to find a solution to the halting problem. Therefore, the nth busy beaver number is larger than any computable number with an n state turing machine.
If we could implement chained arrow notation or up arrow notation in an n state turing machine, then we already know that the nth busy beaver number will be larger than any number possible with chained arrow, or up arrow notation, (or any other notation for large numbers).
So yes, in a way, there are numbers that are finite, but not reference able. (by the way, busy beaver numbers do in fact exist, and they are finite, the first four are 1, 4,6,13, and the next one is at least 47,176,870)