This seems to be a pretty open question to me. Some tries:
First, he asks you about your
definition of a number
Whatever you answer, he may
tell you about some of his thoughts about what qualifies as a number. If $\pm$ infinity qualifies as one, that would work for x.
Then of course there are
number spaces like $\mod 1$ in which e.g. $0.5 \equiv 1.5 \mod 1$
If you don't really have a
definition... I guess the mathematician would love to come up with something for you. For example just define that $x$ and $1$ should be "numbers", and $+$ means just ignoring all numbers after it. Then $x = x + 1$ would be true.
...but that's a pretty unusual thing to do. In real life I think I have already seen this:
$1$ meaning "everything" in set theory and $+$ meaning merging sets (being used instead of $\cup$). Then if $x = 1$ you merge two identical sets which results in the same set $x$. So $x = x + 1$ would be true here. But then sadly I guess sets don't quality as numbers? It fit so well that I wanted to include it though.
Oh, another one:
If you define $=$ as binding stronger than $+$ then technically, the equation $x = x$ is of course true for any number and the $+1$ can just be ignored
All of these approaches seem a bit cheet-y to me. I'd love to see a more elegant solution of this.