TITLE: Is there a number so large that we could never calculate it?
QUESTION [10 upvotes]: Note that I edited this post significantly to make it more clear (as clear as I think I could possibly make it). First, let me mention what I am NOT asking:

I am NOT asking for the largest number we could calculate.
I am NOT asking if there is a largest number (if there were, just add one and it would be bigger, so obviously this is false)
I am NOT asking for something like "divide by zero" or for limits ("infinity" isn't an answer, that isn't a number)


I have been thinking a lot about numbers and I put a couple numbers into Wolfram Alpha just to see what happens. It can handle insanely huge numbers, but it fails as soon as you put in $10! ^ {10!}$. After thinking for a few days, I feel that there must be a largest number that we could possibly ever calculate or define. In fact, I think there are an infinite number of numbers greater than anything we could ever calculate (I am saying larger numbers exist, but we cannot do anything with them, except maybe prove there are numbers so large we can't calculate them). 
I mean a number or an equation that results in an exact number (such as $x!^{x!^{x!^{x!^{x!^{...}}}}}$, where $x$ is an exact number that can be written out) I think there is a limit here. There is a largest number possible because there are only so many atoms in the observable universe. But how would one prove this (that there are numbers so large we couldn't define them)? Is there any information on this? 
Interestingly, of the irrational numbers we use, they always have an alternative way to write them such that the answer is exact. One can simply write the infinitely long irrational number that is the square root of two as $\sqrt 2$. Two symbols and you define the infinitely long number that is the square root of $2$. One can use the formulas for finding $\pi$. But aren't there numbers that we couldn't even do that for? Aren't there numbers that we couldn't write in some form, such as how we write $\sqrt 2$? Such that the number is exact? I think these numbers must exist, but we can't do anything with them. But I have no idea how one could prove that or if there is any information regarding these numbers. So my question is: are there numbers that are just too large (or too precise) that we can't write them down in any way at all that expresses their exact value?

UPDATE:
I accidentally came across something that really helps with my question. I found this article here, and I saw  point #6: 

Unknowable Thing: There are numbers that can’t be computed.
This is another mind bender proved by Alan Turing.

That is exactly what I'm talking about! I am going to do some research on Alan Turing!

REPLY [5 votes]: A necessary condition for being able to calculate a number is that we can specify a computation whose result is that number.
If we accept r.e.s.'s figure that the universe cannot contain more than $10^{185}$ bits, this should mean that
$$ \text{the first number whose Kolmogorov complexity is}\ge 10^{185}+1$$
cannot be produced in any deterministic way, in any effective notation.
(It's not just that if we have that number in our hands, we cannot prove that it answers to the above description -- though that is certainly true too. It's that it's impossible to "have that number" in any meaningful way, even without knowing it satisfies that definition).