When does your nasal bone stop growing?

This is a tough question, and there isn't a clear answer. First it varies widely between people. Long bone growth (arms and legs) and hence the vertical height, is thought to stop about 2 years after menarche, this is the growth spurt of puberty. It is assumed, but not proven, that the facial bones also follow this time table. The average age of the onset of menarche is North America is 12.5 years. Less than 10% start before 11 and 90% have menstruated by 14 years old. So the average age for female noses to finish growing would be 14.5 years with the great majority between 13 and 16 years of age. Boys are later by about 2 years.  

With some variation most females have complete nasal growth in the early teenage years. In males and several years later.

Typically most patients have an adult size nose by the age of 16-18. Females mature a little faster physically than males, hence the 2 year spread. The face around the nose typically stops growing by 18-22 (again with a female vs. male effect). That is why rhinoplasty is not typically done before age 16 in females and 17/18 in males. Each person is unique, so it is possible to be outside those ranges, based on your genetic predisposition

The nose will generally reach 95% its full size by age 18.  No studies have accurately pinpointed an age which you stop growing, but its usually at the end of puberty which can vary in patients.  

Young women's growth matures earlier than young men. As a general rule when you have reached your expected vertical height your nose has stopped as well.  This is often the same time that a young woman begins menstruating. After age 16 I would expect little change.