I haven't had a look at this thread before but I think a few fault finding tips might help people out for when their battery goes flat repeatedly.
Quite obviously something is staying on permanantly and killing the battery. Traditionally this would be because someone left the lights or interior light on. Audis have a timer on the interior lights though. Even if left on the on position (at the switch on the roof) they will still switch off after about 5 minutes. Whether the door is opened or closed or the doors locked or not.
I don't know if there is a similar power cut off for the side and tail lights though.
Where to check
One way to test for a perm power draw is by removing fuses one by one. Some vehicles have a few main (large) fuses that come straight from the pos, one or two of which will then go into another fuse box and will be split off from there. I don't know if it's the same on the A3 though.
If it's the same as this then they are the best fuses to start with, if not then try separating the cables coming straight off the battery and seeing which is drawing power. If this is not possible then you may need to start removing fuses one by one from the fuse boxes.
If you have access the VCDS then it's best to have a look if there are any faults first of course. But bearing in mind that when the ECU detects a fault it will kill the power source I would guess that this type of fault won't be there.
How to check
If you're going to go ahead with the fuse method then the only thing to do is pick a start and end fuse and begin pulling until the draw has gone! It is VERY IMPORTANT that you do not put the fuses back in straight after you have pulled them out. You need to pull them out one by one until the fault goes. Once you have identified the fuse to blame then you can start putting them back in one by one, leaving the problem fuse out. After each 5 you put back in you should check the fault has still gone.
You should carry this out as above just in case you have more than one fault. Or if one circuit depends on another.
How to test for a current draw
This is going to be the most difficult part but if you have read up until now you will see it's the most important.
To test if there is a current draw from the battery you 'simply' need to test what the current flow is with the ignition off using a multimeter. I say 'simply' because you obviously different multimeters can handle different amounts of current flow. I use a Fluke 25 which has a 10amp fuse. This should be more than adequate for this type of job but it's still not a lot! Basically check the current rating of your multimeter and if it's lower than 5 amps I wouldn't go any further.
The need to disconnect the pos lead from the battery and put the ammeter in the circuit to measure the current. The car will always draw a small amount of power and that varies from car to car. I don't know how much an A3 draws but I will test it when I get back to work so I know for future reference. It will be a lot less than an amp though. Apparently some cars will draw a lot of power after switch off for a certain amount of time then it'll drop right down. This just makes things more complex!
You could have any amount of current draw if you have a fault but if it's anywhere near an amp you have definately got a problems and can start with the fuse method.
It is possible to buy a current testing device that plugs into certain multimeters which can handle very high current. It looks like this:
PM me if you want to know how this works!
So anyway I have written this in a rush so sorry if it's a bit all over the place. Hope it can help out a bit. The best tip I can give for someone who's battery keeps dying is to disconnect the battery between uses until you get it fixed. Constant discharge and recharge will ruin your battery.