Why does acidity increase going from sulfide to sulfoxide to sulfone?
The oxidation state of sulphur and its therefore increased inductive effect should be enough to explain this effect.

Besides the number of oxygens, the principle difference between the three structures is the oxidation state of sulphur that rises from $\mathrm{-II}$ to $\pm0$ to $\mathrm{+II}$. The higher an element’s oxidation state, the stronger it draws electrons towards itself since a higher oxidation state typically correlates with less electron density.

Unfortunately, we cannot draw good comparisons. Phosphorus would suggest itself, but neither the Evans table nor the Bordwell data have enough data points for scrutinisation. Furthermore, phosphorus is less electronegative than carbon, meaning the bonds would be polarised towards carbon.

Chlorine might be an option being on the other side of sulphur in the periodic table, but to the best of my knowledge no ‘chlorone’ or ‘chloroxide’ organic compounds are known let alone their acidities.

Nitrogen in its higher oxidation state (a nitro group) has a double bond which can resonate with the negative charge so it is not a good comparison. And the elements in the fourth period and below are again too electropositive in general. If anything, we may be able to draw conclusions based on the &#x3b2;-acidity of an ether, the &#x3b1;-acidity of an acetal/ketal or the &#x3b1;-acidity of an — if these are even stable enough to be deprotonated.