Upper bound on cardinality of a field Is there an upper bound on the cardinality of a field?
The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than continuum?

Yes, There are arbitrarily large fields. This follows from a result in logic, namely the Upward Löwenheim–Skolem theorem (see http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem). 
To be more concrete, consider the countable algebriacly closed field, $\bar{\mathbb{Q}}$. Every element of this field is algebraic (meaning it is a root of a polynomial). However, we can begin adding "transcendental" elements, like $\pi$ and $e$. These don't affect the "field properties" of the field. Then, we can add any cardinality of transcendentals to construct a field of that particular size.    
