In which type of solution does an equilibrium always exist?

You could argue that any solution with water in it has an equilibrium in it of some sort.  If a solution is over saturated there is an equilibrium between the dissociated and solid solute (for example NaCl(s)⇄Na⁺(aq)+Cl⁻(aq) when in water).  Even if the solution is not over saturated, water always has the reaction 2H₂O(l)⇄H₃O⁺(aq)+OH⁻(aq) since water can act as both an acid and a base (this reaction is also always at equilibrium and the equilibrium constant is 1×10⁻¹⁴).

Since we usually ignore the autoionization of water (unless dealing with acid base chemistry), I think the answer your teacher is looking for is over saturated solutions.

I hope this helps.  Let me know if anything is unclear or if you need a different answer.