Is Rayo's number really that big? I was reading about large numbers, and came across Rayo's Number which is defined to be the smallest integer that is not nameable by any expression in the language of set theory that contains less than $10^{100}$ symbols.

Now, my question is: Is this number really that large?

If we pick some "random" number $2091580284...384901284021$ with $10^{100}$ digits, wouldn't it be non-nameable with less than a googol symbols? Wouldn't this number be bigger or equal to Rayo's Number?
As some of the comments already mentioned, you misquoted the definition. The correct definiton (quoting Googology Wiki) is:


  the smallest positive integer bigger than any finite positive integer named by an expression in the language of first order set theory with a googol symbols or less


So while there are only approximately $(10^{100})^{(10^{100})}$ possible expressions, and only a very small fraction of them actually name a number, Rayo's number can be very large. 

We don't exactly know how large, but there are good heuristic arguments that the busy beaver function $\Sigma$ can be implemented in a few million symbols. Given that $\Sigma$ grows much faster then $\mathrm{TREE}$ and all other computable functions, Rayo's number is much larger than  $(10^{100})^{(10^{100})}$ .
No, a number like 10^(10^100) is much smaller than Rayo(10^100), because there are not a Googol digits, but a Googol symbols in first order set-theory, and it is pretty efficient to write down big numbers such as TREE(3), which can be expressed with MUCH LESS than a Googol symbols, you don't need anything like a Googol symbols to express TREE(3),and if we can say that :
              Rayo's Number &gt; TREE(3)

We can certainly say that :
              Rayo's Number &gt; Any number with 10^100 digits

So Rayo's Number is really that big.