Well lets see..

Do you mean challenge two "chatterbots" against eachother, in an argument of sorts??

There is the Turing test, which tends to test a human against a machine, with the tester being blind to which is which.. this isn't so much a competition though, as a benchmark of the AI's abiltiy to mimic human behaviour.

Then theres the always-enjoyable "battle bots" type games, where you have an arena, with a few simulated "bots" which all have the ability to shoot and move, and you have to write the most intelligent AI, that keeps the bot moving out of harms way while inflicting pain on other bots..

There are a few URLs around here for "chatterbot" sites, none come to mind.. for the fighting robots, I think IBM has a Java-based program called Battlebots.. look around a bit, you'll find something.

I've also found a couple more battle-bot style simulations on sourceforge.net..