What you ask for is a bit tricky because you are using an informal concept to describe variables, rather than the very exacting formal definition of what 'x' really means. In general, informal concepts tend to run into trouble when you try to take the opposite of them, because they weren't sufficiently rigorously defined to survive such a treatment.
That being said, there are many cases where we wish to talk about a domain of "everything." Such a thing is often called the "Universe," and it encapsulates every "thing" which one may wish to talk about using the language of mathematics. It is often given the symbol "U," but it is typically referred to using words first, to make its meaning perfectly clear.
This "Universe" does not behave like a normal number. It behaves more like a collection of things. In fact, the mathematical term for it is a "class" (which is in contrast to calling something a "set," which is a easier to understand collection).
From there, you can look at the "for-all" notation:
∀. I can write something like
∀x∈S, which is a series of symbols which means "for all possible values of
x which are elements of the set
S, the following expression is true." This for-all operator has its own behaviors in First Order Logic. You can also write something like
∀x, without the "element of S" specification. If this is done, it is assumed that x is any value within the domain of discourse (i.e. the Universe). This is probably as close to an "opposite of x" as you will get.
Of course, we also have several other meanings of the word "opposite." All of these are other concepts which are associated with the word "opposite," but have very different meanings than the one you appear to be referring to:
-x - additive negation of x
1/x - multiplicative negation of x
¬x - logical negation of x (strong)
not x - modal negation of x (weak)
x* - complex conjugate of x
xᶜ - set theory complement of x
If, in fact, what you were actually after was a symbol for "all numbers" and the whole discussion of "opposites" and "x" was a false path, we do have symbols for that. Most of them are written in the so called "blackboard font":
- ℤ - The set of all natural numbers
- ℝ - The set of all real numbers
- ℚ - The set of all rational numbers
You rarely see a symbol for "numbers" in general because mathematicians don't have much use for such a symbol. The different sets of numbers have different enough properties that mathematicians typically want to be specific as to which set they are referring to.