The Deuce: It does kind of have dual connotation — some use it almost as synonymous with urgent. But literally

it means as soon as you can possibly get it done. If you were on vacation you couldn’t do it until you returned.
I always have/had office staff and patients to put a specific date down if there is a specific deadline it is needed by. Otherwise it gets done when I can get to it.

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We have patients that will drop off forms (parking placard apps, FMLA paperwork, letter requests, etc) at our front desk to the staff.
The staff will take the form and send a patient chart note to us stating the form was dropped off and placed in the doctor's box to complete. Lately, every chart note states: "patient requests form to be completed ASAP"
To me, this means urgent. Not stat, but definitely more timely then routinely done.
Started stressing me out as I felt I was falling behind.
Office Manager then tells me it's just when I can do it. Wasn't this the way I always did it? What is urgent/stat then?