On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite field.)
What is the simplest way to give an example of a field (and an ordered field) of a specific cardinality $\alpha$?
I see there is the "Field" of surreal numbers, but it is a proper class rather than a set (and hence do not have a cardinality as such). However, there seems to be some modified construction which gives proper fields with the cardinality of some strongly inaccessible cardinal.