Let $R$ be a totally ordered set. We can construct a well-ordered cofinal set $S \subset R$ by transfinite induction and the axiom of choice. (Choose an element from $R$ as the base case; then if we've chosen elements $s_i$ for all ordinals $i < \lambda$, if $$R_\lambda = \{ r \in R \; : \; r > s_i \textrm{ for all } i \}$$ is nonempty, pick $s_\lambda \in R_\lambda$. If $R_\lambda$ is empty, we're done.)
Here's my question. Can we, perhaps given some data about $R$ such as its cardinality, put an upper bound on the ordinal corresponding to $S$? Or is it possible for well-ordered cofinal subsets of $\mathbb{R}$, say, to have arbitrarily high ordinality?
You trivially get that the ordinality is lower then the cardinality (and you can realize that bound by just taking the cardinal), but you can also get (from an ordinal view) very big differences. (I'm assuming AC for all of this answer).
Consider that the size of the Reals may be a weakly inaccessible (i.e. big). The reals are totally ordered, but the largest well ordered subset has ordinality strictly smaller then $\omega_1$. This follows since the rationals are dense and so if you had a well ordered subset of size $\omega_1$ it would have a countable cofinal subset.
I can't quickly see if you can get past the $2^\kappa$ boundary though.
Edit: I think there is actually almost certainly an upper bound of distance in terms of the beth function. This follows from some topological considerations I believe. I will check Juhasz when I get home.