2006 chrysler 300 stored for winter. will not

Quick answer is no because you will have no ignition or fuel pump.
It is called anti theft.

You can try out some guide from here reliable-store they also provide live chat feature for instant help and it's very useful

It will need reprogrammed with a scan tool

If a car goes into limp mode, its likely something spectacular has gone wrong and the car is protecting itself from major damage. You will probably have to go to a Chrysler Dealer to rectify and get them to clear the fault off the cars computer.

Have you connected the earth wire on starter?

have a load test done on the battery as it may have a dead cell. Check that the battery posts and terminals are shiny clean and tight.

if the starter relay is clicking that means the starter is bad .

It is all about troubleshooting. Possible causes are a dead battery. Did someone try to steal it and cut the wrong wires underneath?
The starter solenoid could be bad.
If the headlights stay bright when you try to start the car then it is not the battery. if you hear clicking and the headlights are very dim then it is likely to be the battery. If you have full battery with good cable connections and you get very active clicking but no starts then the solenoid on the starter has burned contacts. if battery is good but no clicking when trying to start then the problem is in the key area or alarm system.
All the areas mentioned above are place to start and troubleshooting steps.

could be associated with starter, also could be battery cables, check for looseness or white color on cables.

I had exactly the same problem this morning after returning from the store. I replaced the OEM battery with an Optima Red Top Model 25. Problem gone. Chrysler OEM batteries rarely last more than three years. You don't need to buy an Optima, but they do last a long time.

Good luck....