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Conveniently, the online CRC Handbook provides a table of vapor pressures of saturated aqueous salt solutions $[1]$. To find the one that would give the highest osmotic pressure, we want the want the one that yields the greatest vapor pressure lowering, i.e., the one with the lowest vapor pressure at a given temperature. Of the 19 listed salts, the one whose saturated solution has the lowest vapor pressure is, by a large margin, $\ce{LiCl(aq)}$. The vapor pressure lowering increases with temperature (not surprising, since most salts become more soluble with temperature). Unfortunately, the table stops at $\pu{25 ^{\circ}C}$ for $\ce{LiCl(aq)}$, so that's what we'll use.

Conveniently, the online CRC Handbook provides a table of vapor pressures of saturated aqueous salt solutions $[1]$. To find the one that would give the highest osmotic pressure, we want the want the one that yields the greatest vapor pressure lowering, i.e., the one with the lowest vapor pressure at a given temperature. Of the 19 listed salts, the one whose saturated solution has the lowest vapor pressure at $\pu{25 ^{\circ}C}$ is, by a large margin, $\ce{LiCl(aq)}$.

$$\bf{\Pi = \pu{2985 \bf{bar}}}$$

[$\pu{1 bar} \approx \pu{1 atm}$]

$$\bf{\Pi = \pu{3,000\bf{bar}}=\pu{2,900 \bf{atm }}=\pu{43,000 \bf{psi}} }$$

[I've rounded to 2 significant figures because $k$ has two sig figs, and, as explained above, assuming it's pressure-independent is itself an approximation.]

Indeed, if I were to extrapolate the vapor pressure lowering of $\ce{LiCl}$ to higher temperatures, I could calculate even higher osmotic pressures.

Having said that, if technology were not a limit, we could achieve even higher osmotic pressures than those available from saturated aqueous solutions by using a mixture of water with some liquid that is miscible with water in all proportions. One example would be ethanol. If you separated pure ethanol and pure water with a membrane permeable to water but not ethanol (and such membranes do exist—they are used by distilleries), the initial calculated osmotic pressure would be infinite, because the mole fraction of water on the ethanol side would be $0$, and thus the chemical potential of the water on the ethanol side would be $-\infty$.

Furthermore, I could calculate an even higher osmotic pressure by extrapolating the vapor pressure lowering of $\ce{LiCl}$ to a higher temperature.

Having said that, if technology were not a limit, we could achieve even higher osmotic pressures than those available from dissolving salts in water, by instead using a mixture of water with some liquid that is miscible with water in all proportions. One example would be ethanol. If you separated pure ethanol and pure water with a membrane permeable to water but not ethanol (and such membranes do exist—they are used by distilleries), the initial calculated osmotic pressure would be infinite, because the mole fraction of water on the ethanol side would be $0$, and thus the chemical potential of the water on the ethanol side would be $-\infty$.