This answer shows that the Dedekind completion of the set of hyperreal numbers, endowed with the usual definitions of addition and multiplication of Dedekind cuts, is not an ordered field. But my question is, is it possible to embed this set with its addition and multiplication operations into some ordered field?
I imagine you’d need to add an awful lot of numbers which are greater than all the natural numbers but less than the infinite hyperreal numbers.
No: addition is not cancellative, so it cannot embed in any abelian group. For instance, if $C$ is the Dedekind cut consisting of all elements that are less than some integer, then it is easy to see that $C+1=C$.