What is the minimum mass of an object in space before gravity naturally shapes the object into a sphere?

Planets made of solid rocks are never spherical
(as evidenced by mount Everest)

It depends on the makeup of the object, the degree of rotation (the star Vega is nearly 20% wider at its equator than across the poles, so it's a very oblate spheroid), and its mass.

There is no real minimum.

Depends on the relationship between the
mass of the object and its ductility.
Water droplets in free fall are spherical at
very small dimensions.

It involved the spin of the object and its material state to a large degree.  Certainly, a molten liquid planet wouldn't tend to have mountian ranges and canyons.  The spin distorts the natural shape to more ovaloid from spheroid.  Asteroids (et al) are composed of solid matter (ice and rock, for instance).  It is less influenced than our planets, which had a molten or gaseous past (and some also present)

I am no expert in this field, nor did I do any research about this but here is my two cents.

 I think the more important question is what kind of mass and how close it is to another mass.  A mass of gas that is far enough from anything else will come together to form a sphere all by itself due to the gravitational attraction between the individual atoms.  I think that this can happen with as little as two atoms.  But at this mass, it is highly susceptible to other gravitational forces.  Even a star a few thousand light years away can affect it.  The more mass, the greater chance that it will form its own sphere.  Also, more mass means it will do so at a quicker rate.  

But if we were to take a chunk of solid rock and magically make it bigger, then I think it would be able to maintain its irregular shape up to a point.  Yes, even though the bigger the object, the stronger the gravitational force it has pulling on itself, but don?t forget that the more mass, the stronger internal structure it has.  But, since all matter began as gas, then yeah, most things would start to form sphere depending on how many outside factors are affecting it.

The mass at which this happens depends on the composition and condition of the body in question. The body has to be massive enough for its gravity to overcome the rigidity of its material. If there's enough force to make the material flow, the object will assume a spherical shape because that's the lowest energy configuration - everything on the surface of a sphere is at the same gravitational potential, so there's no tendency for it to fall to another position.

The asteroid Ceres appears to be the smallest spherical asteroid, with a mass of around 10^21 kg, so that sets an upper bound for the minimum size for a rocky object.