Clearly a field $F$ with no Galois extensions must have only non-separable elements in any extension (otherwise, take the minimal polynomial of some separable element over $F$ - its splitting field will be a Galois extension of $F$). This argument reduces the possible examples to non-perfect fields, or in other words - infinite fields of positive characteristic. However, I can't come up with an example of such a field with no separable elements over it.
Are there any such examples?
EDIT: I should have been clearer - I'm looking for a field that has non-trivial algebraic extensions, but has no non-trivial Galois extensions.