Obviously, given you know how to write the statement "There is no largest real number," you can just use R≤0, which would be
$\lnot \exists x \in \mathbb R_{\le 0}, \forall y \in \mathbb R_{\le 0}, y \leq x$
But if you just wanted to $\mathbb R$, how would you express it? Is $\lnot \exists x \in \mathbb R, \forall y \in \mathbb R, y \leq x < 0$ correct? I think it should be able to be expressed as such, but I am not sure...
The suggested formula is not quite right because it would also fail for $\mathbb Z$. The problem is that it fails because the placement of the $\forall y$ means that you are insisting every real $y$ is smaller than $x$, not just the negative ones. You could fix it by saying $$\neg\exists x\in\mathbb R\, (x<0)\wedge\forall y\in\mathbb R\, (y<0\Rightarrow y\leq x).$$