If there where no friction at all, would a spinning wheel held up by one end of the axis spin precess forever without falling down?
I just asked another question about the same problem:
Direction of torque precession of a spinning wheel
Since it seems to be a good practice on stackexchange not to ask several questions in one post, I splitted them up into two questions. However if I am wrong, feel free to merge this questions.
Best Answer
It is spinning forever. As you see, change of angular momentum
$$\frac{\text{d}\vec{L}}{\text{d}t} = \vec{\tau}$$
is always perpendicular to angular momentum itself, which means that angular momentum's direction is changed, while its magnitude is constant. Note the mathematical analogy with velocity and acceleration in case of circular rotation with constant velocity:
$$\frac{\text{d}\vec{v}}{\text{d}t} = \vec{a}_\text{cp}$$