An ionic compound dissociates into 3 ions of 3 different elements?
Yup, they are definitely possible. Such salts are called as mixed salts. They furnish more than two ions when dissolved in water. A popular example is Mohr's salt, popularly used as a standard titrant to measure concentrations of oxidizing agents. It's (anhydrous) formula is:

$$\ce{Fe^{II}(NH4)2(SO4)2}$$

This salt on dissolving in water would split as:

$$\ce{Fe^{II}(NH4)2(SO4)2 &lt;=&gt;&gt; Fe^2+ + 2NH4+ + 2SO4^2-}$$

This is one such example. There are many others, notably alums. Alums are a general type of mixed salts, of the form:

$$\ce{A^{I}M^{III}(SO4)2}$$

where $\ce{A^{I}}$ is a monovalent metal ion, and $\ce{M^{III}}$ is a trivalent metal ion. Clearly, these are also mixed salts.

These are just a few practical examples. There's literally quite nothing that can stop imagination of course.