Metrics w.r.t which $\mathbb R$ is not complete I have seen some examples of metrics on $\mathbb R$ which make it incomplete, like $d(x,y)=|e^{-x}-e^{-y}|$ and $d(x,y)=\arctan x - \arctan y$ where $1,2,3,\ldots$ is a Cauchy sequence which doesn't converge. I am curious about what other examples there are. Is it always infinity which turns out to be 'missing' or are there metrics with Cauchy sequences that do not converge but remain bounded?
That depends on how much leeway one has in specifying the metric.  Consider for example the fact that there exists a bijective map $\phi\colon\mathbb R \to \mathbb R \setminus \mathbb Q$.  If you define the metric on $\mathbb R$ by setting $d(x,y)=|\phi(x)-\phi(y)|$, it will be a metric alright, and clearly not complete.
Cauchy sequences always remain bounded:

If $\epsilon = 1$, there is $N$ such that for all $n,m&gt;N$
$$d(x_n,x_m)&lt;1$$

Therefore, $d(x_n,0)\leq d(0,x_{N+1})+d(x_{N+1},x_n)&lt;d(0,x_{N+1})+1$, for $n&gt;N$ and a fixed. This means that the whole sequence is inside the ball with center $0$ and radius $\max(d(0,x_{N+1})+1, d(x_1,0),d(x_1,0),...,d(x_N,0))$.

In your examples, $1,2,3,...$ is bounded.