What is the highest the boiling point of water can be raised?
The "particular constant" $k$ you refer to in the question isn't an arbitrary constant.  This "law" is usually written as

$\Delta T = km = \frac{M_1 RT_b^2}{1000 \Delta H_1} m$

where k is the "constant you refer to; $M_1$ is the molecular weight of the solvent, $R$ is the gas constant, $T_b$ is the boiling point of the pure solvent, the constant 1000 makes the formula applicable only to water, and $\Delta H_1$ is the heat of vaporization of the solvent, here assumed to be water.  $m$ is the molality of the solute.

This equation is derived in courses in physical chemistry and is itself a simplification to allow slide rule simplification (this is true for a lot of formulas in chemistry).

So what you want to do to make $\Delta T$ as large as possible is to find a solvent with a large molecular weight and a high boiling point and a low heat of vaporization.  The constant 1000, which is the number of grams of solvent in a liter of solvent will also have to be adjusted.  And of course $m$ can be played with.

I don't think anyone has ever worked out what works best for a large $\Delta T$ since boiling point elevation, once used for molecular weight approximation, isn't used for that purpose any more -- and thermometers are a lot more sensitive and accurate than they used to be.