Some ideas. In a Roadster, anytime you turn the key on, if a door or the trunk lid is not fully closed OR if the top is not in it's fully up or fully down position, (or if the top module THINKS its not fully up or down), you will get a chime, warning you of that.

In the Coupe, I can't think of anything other than the driver door sense switch that would cause this. That switch (and the passenger door and trunk lid switches) are wired directly to the CLP-SSM (Central Locking Pump-Security System Module), the CLP-SSM then sends a message over the CAN BUS to the BCM. zip439 says to disconnect the battery to reset things, this may be good advice, perhaps either the BCM or CLP-SSM have "glitched" and got 'stuck' thinking the driver's door is open.

(In my SE, only the driver's door causes the chime, I don't have an SRT or Limited to test, I assume that all Crossfires only chime when the driver's door is opened, I don't know that is the case for sure, but I'm pretty sure they are all alike. Perhaps your issue is the passenger's door switch or the trunklid switch, I doubt they could cause this based on how my car acts, but perhaps another owner can check and chime in.)

Now since this all comes from the CLP-SSM, and since we all know how suseptible to water intrusion that device is, I suggest you remove the trim from the trunk and dig out the CLP-SSM and check for water intrusion. Again, I've never owned a coupe, I understand the CLP-SSM is under the trim in the trunk, in Roadsters, it's in the passenger side rear wheel well, but again, I don't think that's where it is in a coupe.