Precession alone does not involve changing of the angle of the axial tilt, only it's orientation. Since the tilt is 23.439°, that is the radius of the circle upon the celestial sphere of the precessional circle. In about 12,000 years the axis will be relatively close to Vega, which from my latitude can appear nearly straight up at certain times of the year.

Of course, there are other forces acting on the Earth that DO change the axial tilt. For example, Milankovitch cycles.

http://en.wikipedia.org/wiki/Precession
http://en.wikipedia.org/wiki/Milankovitch_cycles

Brian