Let $(k,+,.,0,1,<_k)$ be an ordered field, $| . |_k = x \mapsto \max(x,-x)$.
If $\lambda$ is an ordinal, you can define convergent $\lambda$-sequences: maps $f: \lambda \rightarrow k$ satisfying $\forall \varepsilon >_k 0(\exists \alpha \in \lambda(\forall \beta \ni \alpha(|f(\beta) - x|_k <_k \varepsilon)))$ for some $x \in k$. (you can also define Cauchy $\lambda$-sequences in a similar way)
If $cf(k)$ is the least order type of cofinal well ordered subsets of $k$ (or equivalently the least cardinal of cofinal subsets of $k$), then you get the nice property that $cf(k)$ is the least ordinal such that every bounded monotone $cf(k)$-sequence is a Cauchy $cf(k)$-sequence whenever cf(k) $> \aleph_0$. I can't find a way to do without this hypothesis but my method might not be the right one.
edit: This is actually false (see Eric Wofsey's answer).
This means that $cf(k)$ is the least ordinal such that $cf(k)$-sequentially closed sets are closed sets with respect to the order topology. Also, $k$ is complete (there is no dense embedding of $k$ into a proper linearly ordered field) iff its Cauchy $cf(k)$-sequences are convergent, but this result I can prove without assuming that $cf(k)$ is uncountable.
In the case when $cf(k) = \omega_0$, proving that bounded monotone sequences are Cauchy sequences is the same as proving that $k$ is archimedian. So my question is: are there non archimedian linearly ordered fields with countable cofinality? Do we even know the cofinality of some famous non archimedian fields such as fields of hyperreals? (without assuming CH)