There's an interview somewhere with Shane Carruth (the writer/director of the movie Primer, who also consulted on the time travel sequences in Looper) where he admits that time travel results in logical paradoxes that cannot be effectively rendered in traditional narrative format. In Primer we have characters miraculously appearing and going into catatonic states; Carruth explains these as the only way he can think of to try and grasp the paradoxical effects of time travel on a human subject. Essentially, people's minds stop working the way they're "supposed to."
I think that with Looper we have to concede a bit to the narrative's attempt at representation; the way I personally see it is that if we understand our place (as an audience) as being contemporaneous with the actual characters and events happening within the movie's diegetic frame (i.e. setting, but more specific - the time, place, environment, specifics of what is represented), then we might say that after the film concludes, logically the protagonist no longer exists, making it impossible to conceive of the film's narrative "loop"; but if that character was actually caught in a kind of loop, repeating the same events over and over, then we might admit that we simply watched the final revolution of the chain. I agree that it logically runs into some problems; but in my opinion, that's what makes it so much fun.