Battery problem

You didn't state the age of the battery --If 3-4 years replace it

I assume they are qualified mechanics? If so they should have a load tester to test the battery. They should also have a tester to test the alternator. How about the alt belt? is it tight and in good shape? If you have run the battery down this many times it is time for a new one anyway...clean battery cables are a must too.

like Rob said get your car checked out, also it is hard on the alternator making it recharge a battery, you should charge it with a charger. Every time a battery fully discharges it looses some of its recharge ability, and wont fully recover 100%. I have purchased several cheap brands of batteries that were garbage after one year. Its most likely a bad battery

Could be a couple of things happening: when you remove the battery cables the alternator will run at full voltage as it tries to charge a battery it sees as having zero voltage. This is dangerous as the open-loop output from an alternator can reach 16volts. This can but not allways damamge the on-board electronics. Having said that if the computer has not closed down, the alternator should still allow the car to run. If the alt has none or little output the car will stall.

I would check your alternator maximum charge rate is correct, and make sure your cables are making proper contact with the battery posts and the -ve cable with the engine block

Jason

At some point, when a battery gets discharged, it can no longer hold an adequate charge. It just depends on the age and condition of the battery. If you could always re-charge a dead battery, they would last forever. They don't.

Modern so-called "maintenance free" batteries can be more easily damaged by discharge than older designs. Plus, many maintenance free batteries actually need maintenance (more water) from time to time, even though their owners assume no maintenance is necessary. This causes even more problems for modern batteries.

I would check the voltage at the battery when the motor is running. It should be slightly above 14 volts. Then load the alternator by turning on headlights etc and see that the voltage stays around the same. The alternator should be able to run the car, lights, etc and have sufficient left over to keep the voltage above 14 volts. If the voltage drops then check the alternator fuse (bolted in) and the voltage leaving the alternator. If low at the alternator then the alternator needs looking at.