I can't find a reference for this, so hopefully this isn't too trivial.
By the classification of surfaces by Kodaira dimension, elliptic surfaces have $\kappa = 1$ while K3 surfaces fall into the category of $\kappa = 0$. However, elliptic K3 surfaces exist. What would the Kodaira dimension be for such a variety?
I'm not sure if there is a distinction in the definition of an elliptic K3 compared to a general elliptic surface --- i.e., they are both defined to have an elliptic fibration. An elliptic K3 doesn't seem to be "more" elliptic than it is a K3 surface, and conversely.