Is a perfectly circular orbit possible? Let's assume that there is a perfectly spherical planet and there is a moon which is also perfectly spherical. Let's assume that there is no atmospheric drag and no other gravitational pull. If the moon is put in a perfectly circular orbit around the planet somehow, would the moon eventually "fall" towards the planet and form an elliptical orbit or would it continue to follow the perfectly circular orbit?

Edit: What i actually wanted to ask is that would the gravity of the planet cause the moon to "fall" towards the planet or would the gravity allow the moon to continue its orbit without bending its path any further towards the planet. I know that no planet can be a true sphere or a cube due to the shapes of the particles.
"Perfectly" is a funny word.

Perfect circles are a mathematical abstraction. Real objects are not "perfect".  So supposing a "perfectly spherical planet" is to suppose something that does not and could not exist. All real planets are made of atoms and anything made of little clumps of matter cannot be perfectly spherical.  Even if you built a planet that was as spherical as possible, it would be distorted by its rotation and the tides. So there are no perfectly spherical planets.

Now you say "put in a perfectly circular orbit". This is like drawing a line that is exactly $\pi$cm long. Again you are supposing something that does not, and could not exist.

What we can do is consider a mathematical model of gravity. If you model the sun and the planet as "particles" (ie point masses) and you model gravity with Newton's law of universal gravity, and if you give the model the system with the exact amount of energy to give a perfect circle, then the system will remain in a perfect circle, it will never become elliptical.

If you use general relativity to model gravity, then the release of gravitational radiation will mean that no circular orbits are possible, all orbits will spiral inwards, however it would not become elliptical. Something similar will happen with quantum models of gravity. 

So your question can only be answered in the context of a mathematical model of gravity.