Metric "in the limit"? Let's say that a function $d:S\times S\to [0,\infty)$ for a countable set $S$ is a metric in the limit if
$$d(x,y)\le \liminf_{n\to\infty} d(x,z_n)+d(z_n,y),$$
$$\lim_{n\to\infty} d(z_n,z_n)=0, \quad\text{and}\quad
\lim_{n\to\infty} d(x,z_n)- d(z_n,x)=0$$
for any sequence $z_n$ that goes to infinity in the sense that $(\forall x)(\exists n_0)(\forall n\ge n_0)(z_n\ne x)$.
We could require that $d(x,y)=0\implies x=y$ as well.
I'm looking for an answer to any one of these:

*

*Is this a familiar idea?

*Does some of the theory of metric spaces still hold for such functions?

*Perhaps there exists a "better" variant of this idea?

 A: This is not a complete answer, but was too long for a comment. I don't know of any place where this exact idea has been studied. However, the notion of metrics at infinity play an important role in the analysis of $\delta$-hyperbolic spaces, so this might be a good model for the idea. In particular, the Gromov boundary [1] of a $\delta$-hyperbolic space inherits a metric structure which is in some sense the "metric in the limit."
A $\delta$-hyperbolic $(M,d)$ space is a metric space, but I suspect that many of the properties of the Gromov boundary can be recovered if you weaken the assumptions on $d$ to only involve metricity "in the limit."
In order for this to work, the definition for a "metric in the limit" is really going to have to reflect the fact that the sequences are "going to infinity" and don't converge to anything. You also want to gain metricity in the limit, and I'm not sure your first definition is strong enough to do that.
As such, my proposal would be to add the assumption that
$$ \liminf_{n\to\infty} d(x_n,z_n)+d(z_n,y_n)-d(x_n,y_n) \geq 0,$$
for any sequences $x_n,y_n, z_n$ which go to infinity. I would also suggest changing the definition of "going to infinity" to be the following:
There exists an $\epsilon$ so that for all $x \in S$, there exists an $n_0$ so that for all $n\ge n_0$, $d(z_n, x)>\epsilon$.
On their own, these definitions are not enough to recover any of the properties of the Gromov boundary. However, with a rough version of $\delta$-hyperbolicity and an extra assumption so that "going to infinity" means that the $d(x,z_n) \to \infty$, I suspect that you might be in business.
[1] Väisälä, Jussi, Gromov hyperbolic spaces, Expo. Math. 23, No. 3, 187-231 (2005). ZBL1087.53039.
