[Atomic-S (OP)]

If I wanted to fill a vessel at room temperature and pressure with four mutually immiscible liquids that would separate themselves out into four layers after having been shaken, are there any chemicals that would do the job?

[Bored chemist]

I can do better: 5 or six, depending on the weather.
Mercury
(galium on a warm day)
Perfluorodecane
Saturated salt solution in water
Acetonitrile
cyclohexane.

[evan_au]

Slightly less toxic, some bars can serve liqueurs that have different densities and colors, separated into layers.
But you do have to be careful pouring them, as they will mix together if stirred.

[chiralSPO]

I can do better: 5 or six, depending on the weather.
Mercury
(galium on a warm day)
Perfluorodecane
Saturated salt solution in water
Acetonitrile
cyclohexane.

I believe that silicone oil (polydimethylsiloxane) is immiscible with all of the above liquids, bringing our number up to 7...

[Kryptid]

It might be possible to use an ionic liquid (such as BMIM-PF6) in there somewhere as well. Although water is good at dissolving ionic compounds, there is typically a solubility limit. That being said, a water solution that is saturated with the dissolved ionic liquid should be able to coexist with a layer of undissolved ionic liquid.

[Bored chemist]

Slightly less toxic, some bars can serve liqueurs that have different densities and colors, separated into layers.
But you do have to be careful pouring them, as they will mix together if stirred.

Likewise, water is also less toxic and fails to meet the criteria of the OP.

[Atomic-S (OP)]

It appears that solubility is a more complex problem than might at first seem to be the case by simply dividing substances into polar vs. nonpolar.

[chiralSPO]

It appears that solubility is a more complex problem than might at first seem to be the case by simply dividing substances into polar vs. nonpolar.

Yes. There are all sorts of parameters at play. For instance, even polarity is more complicated than it appears: polar just means "has a dipole moment ≠ 0", but there are also quadrupole moments, octapole moments, hexadecapole moments... Then there is the notion of polarizability: molecules are made up of many charged particles which can move around a little, so even perfectly nonpolar molecules will distort somewhat into a polarized molecule in response to nearby electric fields (polar molecules).  Nonpolar molecules like H₂ are not very polarizable, because they don't have many electrons, and those that they do have are in very tightly localized orbitals, while nonpolar molecules like I₂ are very polarizable, with many electrons and diffuse orbitals. Hydrogen bonding is an extreme example of local electric fields within molecules interacting with one another--identifying number of H-bond donors and acceptors of a solvent and solute is very important to predicting solubility.

Then there are more subtle effects like pi-pi stacking, dielectric constant, halogen bonding, van der Waals forces...