This term was used by my math professor when he was teaching us limits. This term also appears online like this "Let f be a function which is defined on some open interval containing $a$ except possibly at $x = a$". Does this mean that $x$ can be equal to $a$ and not be equal to $a$? This term has really been bugging me.
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In the case you quote, it means that $f$ might not be defined at $a$. It could be defined at $a$, but it doesn't have to be. More generally, you're indicating an exceptional case where a hypothesis or a result need not hold. Again, it might hold in the case designated by "except possibly", but it doesn't have to.
Saying that something A happens except possibly for B means that whenever B is not true A happens. In this case B is $x=a$ and $A$ is $f$ being defined.
Usually we say that to acknowledge that a problem may occur, but the situation B is one that we can ignore and does not matter for the actual proof or definition.
For example, we can say that if $a_n$ is a sequence converging to $0$ then $|a_n|<1$ for all but finitely many $n$'s. In this case A is $|a_n|<1$ and B is "finitely many cases". This fact will not change anything substantial about the sequence $a_n$, because we care (in some contexts anyway) about what happens at the limit, rather than the start of the sequence.
It means that your function $f(x)$ can be defined for some $x=a$, i.e. $a$ is in its domain and $f(a)$ has a clearly defined value.
But it doesn't have to, because as you are only approaching that value and never actually evaluating the function at it, it doesn't have to be defined there at all in order for you to take a limit.
Although I tried to give you an intuitive picture, it also makes sense when you look at the formal epsilon-delta definitions.