Does there exist two distinct set which are not an element of a given infinite set? Let $X$ be an infinite set.
Under ZFC, it is true that $X\notin X$.
So there is at least one set which is not an element of $X$.
Is there another element distinct from $X$ which is not a member of $X$?
 A: The power set of $X$ has greater cardinality than $X$, therefore most of its elements are not elements of $X$.
For another answer, since you're assuming the full ZFC and thus the axiom of foundation, $\{X, a\}$ is never an element of $X$.
A: Of course. $\{X\}$, and $\mathcal P(X)$ and $\{\{X\}\}$. And many many more.
Note that $\sf ZFC$ proves there is a rank function defined by $\operatorname{rank}(X)=\sup\{\operatorname{rank}(Y)+1\mid Y\in X\}$. So every $Y$ such that $\operatorname{rank}(Y)>\operatorname{rank}(X)$ must be different from $X$ itself, and it cannot be a member of $X$.
And $\sf ZFC$ proves that for every ordinal $\alpha$, there is a set of rank $\alpha$. So "most" sets are not elements of $X$.

But even more naively than that. Even without the axiom of regularity. Since we can prove that the collection of all sets is not a set, the collection $\{Y\mid Y\notin X\}$ is not a set (since the union of two sets is a set).
So the axiom of regularity is not even needed to show that most sets are not elements of $X$.
A: I'm a bit confused by the question, because it seems to be shifting levels of analysis:
You could be asking a very general question, in which case, given any particular infinite set (e.g., integers, all possible combination of letters, etc.), it would be obvious that one could construct sets that were not contained in that particular infinite set (e.g., the set of integers does not contain the set of presidential candidates from the last election cycle). 
Or you might be asking a very specific question, about whether or not set X lacks elements that one would intuitively think should be in. If set X is defined as containing all elements X1, X2, etc., then there is no element of that type that it does not contain, but it lacks all sets with more than one element. 
We can nit pick about whether X1 (a value) differs from [X1] (the set containing only one element, of that value)... but I suspect that is a distraction here, because if those are different types, then X contains no sets, and your question is again pretty obvious. 
Probably what you are doing is defining your starting set as also containing all possible combinations of those elements, then the set contains those combinations as well. We might introduce a formality to specify that the set does not contain itself, but that is just part of setting up our initial game rules. 
That formality was introduced to avoid a very specific problem, and so it contains no other exceptions. http://en.wikipedia.org/wiki/Russell%27s_paradox
