Is acetone saturated or not?
Typcially, the terms saturated and unsaturated only apply to the carbon chain in itself. A saturated compound would be one whose systematic name ends in -ane while unsaturated ones would end in -ene or -yne — grossly simplifying. Going by this rather simplistically strict definition, at first sight acetone (propanone) is saturated. But.

You may have heard of the keto-enol tautomery. That is, a ketone is always in equilibrium with its corresponding enol form, for acetone that would be propen-2-ol — an unsaturated compound. It is shown in the scheme below. The first indication of the distinction being difficult.



Furthermore, acetone can be reduced by molecular hydrogen to isopropanol. That means that given the correct conditions, it can basically react much like an unsaturated compound would. It also reacts with elemental bromine similarly to how an unsaturated hydrocarbon would, although only the first mechanistic step is identical; only one carbon is brominated in the product.

There is also the measure of double bond equivalents, that is often used in the process of structure elucitaion. Taking the molecular formula of a compound (which can be deduced well from high-resolution mass spectrometry) and performing a simple calculation allows one to arrive at a number of double bond equivalents for a compound. The formula is:

$$\tag{1} \text{DBE} = \frac{2 \ce{C} - \ce{H} + \ce{N} + 1}{2}$$

Where $\ce{C}$ is the number of carbon atoms (including other tetravalent atoms such as silicon), $\ce{H}$ is the number of hydrogen atoms (including other monovalent atoms such as chlorine) and $\ce{N}$ is the number of nitrogen atoms (including other trivalent atoms such as phosphorus).

However, having 1 double bond equivalent or more does not automatically mean that a molecule is unsaturated, no matter by which definition. Take cyclohexane, $\ce{C6H12}$: It has one double bond equivalent which is caused by the fact that it is cyclic. It will not react with hydrogen or with bromine and no sane chemist would call it unsaturated. So this measure is not of much help.



Concluding, I can only say that both sides of the discussion are wrong. It is wrong that acetone will not react with hydrogen to give isopropanol. It is equally wrong that that makes acetone an unsaturated compound. The concept of unsaturation simply does not extend well to $\ce{C=O}$ double bonds.

The exam question should not be graded.