There are many answers to this question. If you count the drones, as F. Ray has done, Bagpipes can have a range of several octaves. But it must be borne in mind that the drones play only fixed, continuous notes, so they do not contribute to the range of melody notes.
Counting only the chanter (the pipe on which the melody is played), it depends on the type of bagpipes. The Great Highland Pipes (i.e. the familiar Scottish bagpipes) - and the related Border Pipes and Scottish Smallpipes - have a range of 9 notes or an octave and one note. Various other kinds of bagpipes exist with a similar range (although not necessarily the same scale or pitch).
Other bagpipes, such as the Irish Uilleann Pipes, Galician Gaita and Northumbrian Smallpipes, have a chanter that plays a full two octaves; Northumbrian Smallpipes can be made with a range of more than two octaves.
Whilst most bagpipes are tied to a particular scale, some also have chromatic capabilities: The Galician Gaita, using a combination of half-hole fingerings and cross-fingerings; The Northumbrian Smallpipes and Uilleann Pipes, using a system of keys.