At 7am yesterday the strange buzzing noise finally got me out of bed
to investigate. The annoying "chime" that bleats when someone presses
the doorbell was buzzing constantly. I headed down to the basement and
found that another bell, attached to the main fuse panel by some
square metal things and some ancient looking wires, was also ringing
constantly. I actually had never really noticed it before, nor even
knew that there was a doorbell ringer in the basement, but it does
make sense.

Anyway, I unscrewed a wire from the downstairs ringer and happily
sighed as the noise stopped both upstairs and down. I then went to the
door and unscrewed the doorbell noting that the built in little light
was no longer lit, checked the wiring since it had been raining and
snowing a good deal, and it looked ok so I screwed it back in. I
hooked up the disconnected wire in the basement and no bells went off.

Somewhere further back in the basement is an odd construction hanging
from the ceiling that looks like a transformer made from square Tower
of Hanoi puzzle parts and which I always thought was part of the
doorbell system. It looks very much like the one hanging from the fuse
panel leading to the ringer but is much bigger.

I guess those are converting house current to DC for the doorbell, but
why there are two, and whether that big one is still functioning is
something that I'll need to test.

Anyway, it seems to me that the doorbell must have shorted and maybe
burned out the little bulb, and set off the bells constantly. Then, I
fiddled with the doorbell and it went back to happy mode and
everything is working ok. One oddity is that a few times I've pressed
it since then and both bells ring, but sometimes only the basement one
rings. I think perhaps the constant ringing has managed to weaken the
upstairs bell. I always hated that one anyway so maybe I should
replace it.

Does this make sense? Is there something else I should check? Maybe
hookup a multimeter across the terminals of those transformers?