Last night before bed, I started to cough. Now all day today, I can't stop coughing! No other symptoms besides a ticklish throat. I have asthma & allergies & received a flu shot. What could this be?

Lots of things. Could be asthma, post nasal drip down the back of throat, or a virus, or a way some people present from reflux. If over the counter cough medicines don't help and keeping a cool mist humidifier by your bed, see your doctor(assuming you tried using your asthma medication ).
Maybe parainfluenza. Virus called parainfluenza is now frequently causing horrible cough that can persist for several weeks. You usually cough all day. Same virus causes croup in young children. Should see physician for exam and possible medication to subdue cough. I personally have had this virus recently and know of many other cases. May need chest radiograph since have history of asthma to make sure lungs are clear.