1.Referring to something that needs immediate action or aid; urgent: The questions asked by the police were of an exigent nature and had to be answered quickly.
2. Concerning someone making heavy demands: The protesters were making exigent or pressing requests of the politicians of their town.

At the doctor's office there was a most exigent patient wanting to see Dr. Smith at once!

Exigent and pressing pertains to an urgency that requires prompt action.