Do perfect spheres exist in nature? Often in physics, Objects are approximated as spherical. However do any perfectly spherical objects actually exist in nature?
 A: If we are considering the states of matter we are familiar with (fermions), since these structures are inherently discretized (solids, liquids, gases), they will not exhibit perfect spherical symmetry. 
We can loosen the definition of perfect and construct a cutoff where variations in the radius are negligible below some length scale and call that perfect.
Replying to wouter, the black hole will have an event horizon that is spherical if and only if the black hole has zero angular momentum.  It ultimately depends on the interpretation of Cactus' question about what an "object" constitutes and if an event horizon is considered a valid answer.  
A: I suppose it depends on how perfect your perfect sphere is.
A drop of water in space with no other gravity effects would look spherical, but if you zoomed in enough to see edges of molecules, then it wouldn't be.
From there you can just keep getting smaller (atoms, protons and neutrons, quarks, etc.) until you get into things like string theorem and quantum foam.
I would say no, like straight lines, there will not be a perfect spherical physical object in the natural world.
A: 
Often in physics, Objects are approximated as spherical. However do any perfectly spherical objects actually exist in nature?

Yes, with a couple of qualifiers. For example, a perfectly isolated 4He atom in its ground state is a perfect sphere according to the standard model of particle physics. This follows because the nucleus is in a spin-zero state, and the electrons also couple to spin zero. In quantum mechanics, a spin of zero is invariant under a rotation, which means that it's a perfect sphere.
There were, however, some qualifiers above. (1) The atom has to be perfectly isolated. In reality, we can't completely shield any region of space from electric, magnetic, and gravitational fields, so at some level these will cause the atom to be distorted. (2) This is according to the standard model, which we know breaks down at some level.
A: No, but it doesn't matter.
The theories that approximate things using spheres are ones in which the final result (the number you measure, the reading on your meter, whatever) depends continuously in some sense on the deviations from sphericity. More symbolically, for any $\varepsilon$ tolerance you allow in your measurement (none of our measurements are infinitely precise), there exists a $\delta$ such that any real object "within $\delta$" of being a sphere will give the same measurement to within $\varepsilon$.
It is not that theories are invalid because they assume something "wrong" about nature. Instead, you have to understand that there is always an implicit statement about how "real" behavior approaches the model as deviations from the model's assumptions get smaller.
A: The other answers highlight the importance of a model vs observations, but we do have plenty of very, very spherical objects, as far as experimental measurements go. I only know of this through the excellent sixty symbols video on the topic, but the electron's distribution of charge has a dipole moment of less than $10^{-28}$ C m, which is pretty dang spherical.
See also: http://www.nature.com/news/2011/110525/full/news.2011.321.html
