How long did it take Dumbledore to grow his beard?

First, you have to figure out how long his beard actually is at a given time. In the link down the bottom, you can see that his head is roughly 1/3 of his total head + beard length. Given that the average head length is about 22.5cm, we could assume that his beard is about 45 - 50cm long from his chin to the tip of the beard.

Now, to figure out how long it took to grow. By doing a quick couple of searches on google, you can see that the average hair growth for an adult male is about 0.5mm per day, which equates to about 18.25cm per year. By dividing his beard length in the photo (about 47cm) by 18.25, we get an answer of roughly 2 and a half years of beard growth.

Keep in mind that these are rough estimates, and most of this information is likely to differ from person to person.