Thanks, XXXXX XXXXX description doesn't fit anything common on the Intrepid line, but it's most definitely something that can be fixed... but it's going to take a little investigation.
I'd start with the battery itself. If it shows signs of acid leakage or deposits on the top of the battery, you may be losing at least part of your charge through electrolytic draw once the acid gets wet. It's a bit of a stretch, but I've seen it happen before.
While the battery is mostly obscured from view, you can see at least the positive post from the air cleaner area and the positive post tends to collect much more acid and form bigger deposits. If the positive battery post looks OK, the negative side probably is as well.
But if it's mounded with crud, it needs to be cleaned up. You have the choice of removing the inner fender access panel (not handy) or working from the top (also not handy). You'd need to lift the car and remove the wheel to come out through the access panel by the way...
From the top, you can remove the air cleaner box and then take the headlamp out for access to the negative side of the battery. The air cleaner box lid pops off with the two large snaps, then remove the push-pin from the right side fender attachment point. The center of the pin is removed to relieve its bite on the fender. A flat-blade screwdriver loosens the large tube at which point the upper section is free.
The lower air cleaner housing then simply pulls off at the rubber grommet on the end of the housing that's closest to the engine.
The headlamp comes out easier than you'd expect... only two bolts, using 7mm hex heads. Unscrew the bolts and pull them out... they're a bit long. The headlamp slips out pretty easily at that point.
A mixture of baking soda and water (1/2 cup soda/ 1/2 gal water) does a great job of neutralizing the acid, but then it needs to be scrubbed a bit and rinsed to get all the stuff off. Consider removing the terminal clamps for cleaning if they haven't been serviced in some time.
It seems likely there may be a circuit that stays awake on your car when it gets wet, something you can confirm by using an ammeter, placed between the battery negative post and the body harness. It's much easier to do at the jump start terminal, where you can remove the battery cable, hang it on the stud using the loop on the cable, then connect your ammeter in series with the car.
A small amount of initial current flow is normal as systems are powered up, but total current flow should settle down to about 60 to 80 milliamps (0.06- 0.08 amp) within ten minutes. Remove the key completely from the ignition for this test and don't open doors during the whole time or the time-out process will be interrupted.
If you see no significant electrical draws, it might be a good time to break out the garden hose. Start wetting parts of the car by working low and going up to make it more obvious what the problem might be if you get a reaction.
No circuits in particular come to mind concerning your problem, but the good news is that you can't hurt anything with the water... so hit everything at least once. Keep an eye on the ammeter for any changes in current flow over time as it may take a while for something to happen.
If you don't already have a digital multimeter, it would be a great time to pick one up. They've come WAY down in price, with decent units available for $20 or less. These units will handle up to 10 amps of current, which is plenty for this test as long as you avoid having the key or other circuits turned on. For key-off draw testing... checks OK. Most meters don't come with aligator clips, so you may need to improvise some way to clip the leads to the connections...
Pulling fuses... one by one... will tell you roughly where the draw is once something starts happening. Some fuses have multiple duties, even feeding smaller fuses inside the car, so it's not always immediately obvious what's causing the problem. If you get a positive reaction, let me know and I'll research the circuit.
Current drain from the top of the battery won't show up on this test, so it's important to have the battery clean and DRY before the test.
A quick and easy test for the battery itself would be to disconnect the negative cable at the jump start post and leave the cable disconnected overnight when you'd expect it to wake up dead the next day. If the battery still dies overnight, the problem isn't with the car... it's the battery itself (electrolytic drain).
If the car still has power in the morning, we'll need to look deeper.
As always, don't hesitate to write back if you have questions or problems. I'll be glad to help!