Ok we have all had our smart arse comments, so down to business. What style of hair drying is it, ie, is it the posh kind that sits above your head, the gun shaped one that has nozzles or the bag on the head with a tube type popular in the 70s?
Firstly check that its not something in any attached bits, if not then if its coming out of the air hole so to speak, it could be that dust or something is inside the unit and getting cooked. As many do use centrifugal fans, these do get very dirty indeed and I suspect this could cause it. Of course taking these things apart is often not trivial if possible. they are made by the same mob who make toasters and seldom let you get in to do much. Mostly the heat is after the fan of course but if the motor is caked in muck then it probably gets quite warm and burns off at the start.
Some laquers used on motor windings can smell a bit of onions, so don't be surprised one day if it just dies due to the motor overheating. With muck in the little places its not supposed to be and rubbish bearings it will eventually succumb. I suspect a thermal cut out will stop it before it catches fire.. grin.
Brian