Let me address merely the suggestion the OP makes in the comment: whether this ordinal can be specified from the cofinality of the field.

The answer is no, because any ordered field $F$ can be elementarily extended to a field with cofinality $\omega$, or indeed, to a field with any given regular cofinality. To have cofinality $\delta$, simply extend $\delta$ many times, making sure to put a new element on top each time, taking unions at limit stages.

$$F\prec F_1\prec F_2\prec\dots\prec F_\alpha\prec\dots\prec F_\delta$$

The resulting field will have the same cofinality as $\delta$. Since the initial field $F$ could have had very large embedded ordinals, this shows that there are fields with very large embedded ordinals, as large as desired, which nevertheless have cofinality $\omega$.

Let me address merely the suggestion the OP makes in the comment: whether this ordinal can be specified from the cofinality of the field.

The answer is no, because any ordered field $F$ can be elementarily extended to a field with cofinality $\omega$, or indeed, to a field with any given regular cofinality. To have cofinality $\delta$, simply extend $\delta$ many times, making sure to put a new element on top each time, taking unions at limit stages.

$$F\prec F_1\prec F_2\prec\dots\prec F_\alpha\prec\dots\prec F_\delta$$

The resulting field will have the same cofinality as $\delta$, because the sequence of those points newly added on top at each stage will be cofinal in $F_\delta$. Since the initial field $F$ could have had very large embedded ordinals, which will still embed into the resulting field $F_\delta$, this shows that there are fields with very large embedded ordinals, as large as desired, which nevertheless have cofinality $\omega$, or any desired cofinality.

Let me address merely the suggestion the OP makes in the comment: whether this ordinal can be specified from the cofinality of the field.

The answer is no, because any ordered field $F$ can be elementarily extended to a field with cofinality $\omega$, or indeed, to a field with any given regular cofinality. To have cofinality $\delta$, simply extend $\delta$ many times, making sure to put a new element on top each time, taking unions at limit stages.

$$F\prec F_1\prec F_2\prec\dots\prec F_\alpha\prec\dots\prec F_\delta$$

In particular, there can be fields with very large embedded ordinals, as large as desired, which still have cofinality $\omega$.

Let me address merely the suggestion the OP makes in the comment: whether this ordinal can be specified from the cofinality of the field.

The answer is no, because any ordered field $F$ can be elementarily extended to a field with cofinality $\omega$, or indeed, to a field with any given regular cofinality. To have cofinality $\delta$, simply extend $\delta$ many times, making sure to put a new element on top each time, taking unions at limit stages.

$$F\prec F_1\prec F_2\prec\dots\prec F_\alpha\prec\dots\prec F_\delta$$

The resulting field will have the same cofinality as $\delta$. Since the initial field $F$ could have had very large embedded ordinals, this shows that there are fields with very large embedded ordinals, as large as desired, which nevertheless have cofinality $\omega$.