The word "most inexterminable" indicates something that is incredibly difficult or impossible to eradicate or destroy. The antonyms, or opposite words that can be used instead, would be terms that describe something that can be easily removed or destroyed. For example, some antonyms for "most inexterminable" might include "temporary," "manageable," "evanescent," "flimsy," or "weak." All of these words imply that the object or situation in question is not permanent, robust, or difficult to overcome. Rather, it is something that can be dealt with quickly and effectively using the appropriate tools or approach.