You could do cash in hand type work and your boss wouldn't ask you for a NI number, but it is dodgy, and most people WILL ask you for it now. Unless you are self-employed.
Incorrect CheekyChops - there are various reasons why a person born and resident in the UK has not been allocated a NINO. Most people are issued their NI Card prior to their 16th birthday because they are a part of the Child Benefit system.
if you have no nino it is the employers responsibility to get you booked for an appointment to get one. you will be put on emergency tax until you get a nino, which is at a high rate. ( i think its about 33%). if you make a claim to benefits then you will be asked to attend a national insurance number interview a couple of weeks after your new claims interview for benefit. the social security system is run on national insurance numbers to pay benefits accurately.
Just to clear up for cheeky chops- Im british- was born in london but left the uk when I was a baby and lived in Ireland. I didnt have a nino and had to apply for one when I got back to the uk. Only took a few weeks- some stupid questions and that was about it. Handy to have otherwise you dont get taxed right