What is the cardinality of the set of mathematical structures? In mathematical logic, there is a set $X$ that contains all mathematical structures given a certain symbol set $S$. 
E.g. lets say $S=(•,R)$ (a binary operator and a binary relation)
Let an $S$ structure $\mathfrak A$ be a tuple $(A, •,R)$ such that the two symbols have a corresponding relation that is defkned on $A$.
What is the cardinality of $\{\mathfrak A|\mathfrak A \text{ is an $S$ structure} \}$?
My suspicion is that it is huge. Perhaps it doesn't have a cardinality? My suspicion is: for any cardinality, we can construct another set of S structures whose domains have that cardinality, so that the set of such structures must have a cardinality greater than it. Hence the set of all such structures has an "unbounded cardinality"? (???!)

The correct term is "a proper class", which means that this collection is not a set.
And while we can certainly talk about proper classes and to some extent treat them as sets for some rudimentary things (e.g. intersections, unions, products), these are not sets. And one thing that proper classes do not have is cardinality.1
Your idea is correct, but your argument is a bit lacking. Indeed, fixing a language, we can find arbitrarily large structure for that language, which implies that this is a proper class. But there is no "direct connection" between the cardinality of the collection of structure, and the cardinality of a new structure. If $X$ is a countable set, then $X\cup\{X\}$ is also a countable set.
The correct approach, however, would be to argue that if $X$ is a set of $S$-structures, then there is some cardinal $\kappa$ such that no member of $X$ has size $\kappa$. Therefore taking any $S$-structure of size $\kappa$, it will not be in $X$. So no set can exhaust all the $S$-structures.

Footnotes.


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*It is actually possible to define cardinality for proper classes, by talking about the existence of bijections between the classes. But this requires better grasp of axiomatic set theory, and understanding what does it mean for objects to live in the meta-theory and how set theory interacts with its meta-theory. So let's just agree that for now, proper classes do not have cardinality.

