Here's a good one:

1983 Diesel vanagon, stock.

Alternator belt breaks, drive home on battery (on a diesel, you can drive
quite aways, since the only battery drain are the instruments and the
fuel solenoid when the headlights are off). The water pump has it's own
belt, so no big deal.

Replace belt, but the battery will not hold a charge. Measurement says
there is a significant (tens of amps) drain on the battery, even with the
key off.

So, the puzzle is: what was wrong?

;)

Jon