Does the property of additivity for the amounts of different substances hold up for ideal gas law?
The question would be better phrased as which gas contributes most to the total pressure? The contribution is referred to as the partial pressure. The sum of the partial pressures is the total pressure. 

The formula you would use to compute the partial pressure $p_i$ of gas component $i$ is either of the following: 

$$p_i=\chi_i p = \frac{n_i}{n}p = \frac{n_i}{V}RT$$

where $n_i$ is the amount of component $i$ in the gas and $\chi_i$ its mole fraction. So the argument "the gas that takes up the least amount of volume with the greatest amount of particles will exert the greatest pressure" is not quite right. It suffices to say that the gas that contributes the greatest amount of particles will contribute (exert) the greatest pressure. If you mix identical masses of each gas, that will be the gas with lowest molecular mass.