Why isn't thallium triiodide stable?
Thallium triiodide does exist  and it is formulated as $\ce{Tl+I3-}$ and not $\ce{Tl^3+(I^{-})3}$. Thus $\ce{TlI3}$ is a thallium(I) compound and contains the triiodide ion, $\ce{I3^{−}}$. This is confirmed by considering the standard potential which indicate that $\ce{Tl(III)}$ is rapidly reduced to $\ce{Tl(I)}$ by the iodide.

$$\begin{align}
\ce{Tl^3+ + 2e− &amp;-&gt; Tl+} &amp; E^\circ &amp;= +1.252~\mathrm{V} \\
\ce{I2 + 2e- &amp;-&gt; 2I-} &amp; E^\circ &amp;= +0.5355~\mathrm{V} \\ \hline
\ce{Tl^3+ + 2 I- &amp;-&gt; Tl+ + I2} &amp; E^\circ_\mathrm{cell} &amp;= +0.717~\mathrm{V} &gt; 0 \\ \hline
\end{align}$$

(data from Wikipedia).

$\ce{TlI3}$ can be prepared by the evaporation of stoichiometric quantities of $\ce{TlI}$ and $\ce{I2}$ in concentrated aqueous $\ce{HI}$, or by reacting $\ce{TlI}$ with $\ce{I2}$ in ethanol.