Is there always a point with no gravitational acceleration? Assuming that there are no 'point particles' but rather particles have finite size and density, and that the force of gravity is defined simply by Newton's law of gravitation:
$$F_g = -\dfrac{Gm_1m_2}{r^2}\hat{r}$$
must there always be a point in space in which I would experience zero net force? 
Considering small objects, the conjecture seems all right, for instance for a single spherical object of uniform density, an object placed at the centre of the sphere would be subject to 0 net force. For two objects, there will be a 'dead spot' of no net force along the line joining the two objects (if the objects are pretty normally shaped). 
When I asked my friend this problem, he proposed that a solution could perhaps be constructed from an application of the Hairy Ball Theorem, though I'm not too sure how to apply this theorem. I would have posted this in Physics Stack exchange, but by using the approximations that I am using (ignoring relativity and using the Newtonian version) it seems more of a mathematical problem. 
 A: Newtonian gravity is a conservative force, so the gravity field is can always be represented as the gradient of a gravitational potential.
If the source distribution is finite (otherwise we run into weird problems of conditional convergence anyway) we can always normalize the potential such that it is zero at infinity. Assuming that the mass density is globally bounded (which is a bit stronger than just assuming there are no point masses), the potential will be globally bounded too. And the potential is always negative too (there's no way Newton's law can produce a potential that is higher than that of infinity), so it will have a global minimum somewhere.
At this global minimum, the gravitational field (which is just the gradient of the potential) must be zero.
A: The globally bounded case is covered by @Henning. However, even if you consider point masses (which planets and stars almost are), and exclude them from consideration, you can show that there are points with zero gravitational force in space (away from the "point" sources), as soon as you have more than one point mass. How? All singularities are negative-valued and Morse theory tells you that there must be a saddle point between each two minima, so you can even tell how many saddle points there are for a given number of masses (assuming the symmetry is not such that they coincide in higher-order saddles). There are also no local maxima.
This is one of the reasons (at least hints) that a 3-body problem in general leads to chaotic motion: at a saddle point, small changes in input trajectory can lead to divergent output trajectories.
A: The answer depends. If you also allow an infinite universe with infinitely many objects and no upper bound on their density, then you can imagine an infinite array of objects. Say, a light object at the point $(1,0,0)$, a more massive one at $(2,0,0)$, even bigger one at $(3,0,0)$ et cetera. You can arrange their densities to grow in such a way that the gravitational field of the next object will overshadow all the preceding ones.
Of course, such a universe is not physically possible, and "everything would have been pulled to the end of the $x$-axes ages ago".
But if the number of objects is finite, then you can form the sum of their gravitational potentials. This approaches to zero as you go far away. Because there are no point masses, the potential will never go to $-\infty$. Therefore it has a global minimum somewhere (it is a continuous function and we just reduced the study to a compact set). The force of gravity will vanish at that global minimum.
