Expanding on Jan's reference to three-center four-electron bonding:
When the three iodine atoms all come together in a line the overlap among all three atoms gives the combination of $p$ orbitals rendered thusly (source: Wikipedia by Arun Sridharan):

Only the lowest orbital gives a bond but, because it involves overlapping all three atoms instead of just two, it's a stronger bond than just the two-center bond in $\ce{I_2}$. However, the three-center bond is distributed over two linkages instead of one and so each individual linkage is less strongly bonded than an $\ce{I_2}$ molecule.
Which leads to the reverse problem. Why is this a property of iodine? Why don't other halogens do the same thing? In fact the other halogens do form three-atom complexes, as we shall see; but they do so with increasing difficulty as we move to lighter halogens.
What enters here is the fact that you don't just have the trihalide ion. It must be bound to counterions by ionic bonds or to polar solvent molecules by ion-dipole interactions. These additional, nonmolecular electrostatistic interactions, are more favorable with a compact spherical ion consisting of one atom instead of an extended three-atom chain. The nonmolecular interactions with surrounding ions ir dipole thus tend to break up the trihalide ions. With iodine the single atom is still a bit bulky and therefore the nonmolecular interactions have little power to break up the triiodide complex. With smaller-atom halogens the nonmolecular interactions become stronger and more important.
What is the lightest known trihalide ion, then, given the fact of these evil nonmolecular electrostatic interactions? The answer may surprise a few people.
Tribromide ion
This ion is actually fairly commonplace if you know where to look, which could be an organic chem lab. Pyridinium tribromide, $\ce{(C5H6N)^+(Br3)^-}$, is used as a brominating agent for reactive substrates such as phenols; it is easier to handle than liquid bromine. We therefore move on to a bigger, or smaller, challenge:
Trichloride ion
The $\ce{Cl_3^-}$ ion has been obtained from electrochemical studies in suitable solvents such as acetonitrile, for example see here.
Trichloride ion is also obtained as an ionic liquid by adding chlorine to an ordinary chloride salt. These species can be used as chlorinating/oxidizing agents in circumstances where organic solvents would be prone to side reactions with the chlorine. The reference given here includes a list of trichloride ion bearing materials, both solid and liquid.
Trifluoride ion
Surely we have reached our wit's end? This reference reported that trifluoride ion has been observed in a low-temperature noble gas matrix and at low pressure in the gas phase. The same reference also reports that the main contributing structures for this ion involves radical combinations rather than the ion-molecule model we might ordinarily expect for this type of complex.
So at least under suitable conditions we can go all the way to $\ce{F_3^-}$!