Jeff,

There is probably a constant current draw.  You need to troubleshoot it.

In my case, the dome lights were the problem.  But you might pull one
fuse at a time, with the car not running, and put an ammeter in place of
the fuse.  You may find one circuit is on all the time.  Maybe it's the
trunk lights or the glove compartment light.  You won't know until you
make some tests.

Bob

Okay, when you park the car, the alternator light isn't on and the car is
fine.  But a day or two later the battery is dead.

This is the sign that something is pulling current while the car is not
in use.  Disconnect a battery line and put an ammeter in there, and see
how much current is being pulled with the car turned off and the doors
closed.  If it's more than a few mA, pull out fuses until you find the
culprit.
--scott

I said in another post that you could be having trouble with a shorted cell
in the battery,  but the thought just occurred to me -- you could be having
a short in the starter.

I have an old Jeep CJ5, and the clicking thing started. I kept getting the
charger out, and jumper cables, to get it going, but eventually the cables
would not do the trick. Anyway, I finally came to the conclusion that the
starter itself was shorted, and it would cause the starter solenoid to fire
(the gear would be thrown into the flywheel), but the starter motor would
not spin.

I think your more likely suspect is a bad battery and/or a load that is
active when the car is parked, but do not forget the starter itself can
cause the symptom you reported.

Voltage is a function of load. If the voltage is low, the load will not be
driven. If the load is high, it can pull the voltage so low that it can not
drive the load.

On mine, I had a sticky passenger door lock.  If I locked the car and
the passenger side didn't go down fully, it would drain the battery
within a day.