Part of the reason that I3- exists because iodine is a much larger atom than fluorine and thus the central iodine atom has more "room" than a fluorine atom would. (Edit: as deepakalways points out...but it's not that simple)
Secondly, if you draw the lewis structure for I3-, you'll see that the central iodine atom does not obey the octect rule. It has 5 electron domains (10 electrons). This means that its d orbitals are participating in its bonding. Fluroine is only in the second period and since there are no "2d" orbitals, it cannot form the F3- ion. If we use this same logic than both Cl3- and Br3- should exist because the central atoms in those ions would utilize their 3d and 4d orbitals, respectively.
I have read that the Br3- and I3- exist but I have never heard of the Cl3- ion. This is probably similar to F3- in regards to the size of the atom. I think that the chlorine atom is too small for Cl3- to exist. I am not entirely sure though- you'll have to look that one up for me.
Both Br3- and I3- only can exist in solution as they are fairly unstable. They exist in equilibirum when Br2/I2 and Br-/I- are mixed. For example:
I2 (purple) + I- (colorless) <--> I3- (brown)
That is why tincture of iodine is brown because it is a mixture of I2 and KI.