Does I_3^(-) exist?

Does ${I}_{3}^{-}$ exist.........?
Well, the answer is yes, and the Lewis structure distributes $3 \times 7 + 1 = 22$ electrons.......
${\left[I - I - I\right]}^{-}$, going from left to right, there are 7 electrons associated with the leftmost iodine, 3 lone-pairs, and 1 bonding electrons, hence this centre is neutral; the central iodine has 3 non-bonding lone-pairs, and 2 electrons from the $I - I$ bonds, I.e. associated with EIGHT electrons, hence this centre is a formal anion; the rightmost iodine has the same electronic distribution as the leftmost, and this is a formally neutral centre.
Note that when you are next in the lab, try slurrying iodine in water (in which of course it is insoluble!). If you add a little iodide salt (say $K I$, if you can find it), the elemental iodine goes up into solution to form ${I}_{3}^{-}$, and other polyhalides......