Can there exist a motion which cannot be described as a function of time? I know there exist systems for which we cannot solve its differential equation, But I was wondering if there could be a motion that cannot be represented as a differential equation with respect to time.
 A: Simple answer; no.
More Involved Answer; it depends what you mean by "motion", and how that applies to different frameworks of Mechanics. Lets take all Classical Motion, we can sum up all of Classical Mechanics in Hamilton's Principle or the Euler-Lagrange equation, these two are fundamental descriptions of Nature based on the fact there are certain symmetries in space and time (translating, rotating, time shifting does not effect experiment), and as a result of these symmetries of space and time, the behavior of a system is always expressed in terms of a differential equation; the fact that time is jammed in that differential equation is a consequence of conserved quantities in our world (i.e. everything classical MUST be describable by a D.E. in time in our world).
But what about a different world? What if there is some classically describable world with no symmetries? There is no equivalence between frames of reference...well in such a world there would be no conserved quantities of motion, and the very study of mechanics would not make sense. Conserved quantities are what we base our Mechanics on, and if we don't have "constant's of motion", then motion itself is a sort of meaningless concept (since I will see an experiment one way and you will see it another, so how can we describe the motion itself). Of course such a world does not exist, and remarkably wherever physicists go, there just seems to be symmetries waiting for them; this has been an immensely rich source of developing laws in Particle Physics and QFT.
Speaking of Quantum, what happens there? Is our motion really described by Differential Equations even though there is all this weirdness? Well...yes! Quantum Mechanics is no different than Classical Mechanics in its use of symmetries to generate constants of motion to develop D.E. however, the symmetries in quantum mechanics are not symmetries in observables, rather in operators. It is a subtle point; but the commutation relation is a critical component of Quantum Mechanics, and so Q.M. is exactly like C.M. except it is describing constants of motion for operators. That is why Schrodinger's Equation cannot tell you how a particle will evolve with time, but it can tell you how its position and momentum operators will.
For further reading see: Noether Theorem, Symmetries in Mechanics, Poisson Bracket's and Mechanics
A: Newton's second law is a differential equation and applies to any motion. Schroedinger equation too. It's up to you if you know or not the expression of the force or of the potential.
So the question is a little tricky. What do we mean by "can be represented by a differential equation"? Do we need to be able to write down the equation to say "it can be represented"? If not, the answer should be always yes, the motion "can" be represented. If yes, it's more like a subjective question, it depends on our knowledge of the conditions.
