Turned off breaker, power consumption almost cut in half. No idea what it's for
For posterity (and future searches), I'll try to combine the suggestions from comments and briefly summarize the outcome. Any edits are welcome, especially if you feel I've misconstrued ideas. This is tagged as a community wiki.


The first suggestion was to leave the breaker off until the issue was better understood, as this could have been a dangerous and well hidden electrical fault.
Being winter, pipe or roof heating was suspected. (We later learned
that the OP is in a mild climate, with overnight temperatures a bit
below freezing at this time of year.)
Suggestion to 'leave it off' and start hunting for things that don't
work. (Particular reminder to check fridges/freezers after turning
off circuit breakers. Nobody likes a freezer full of spoiled food.)
Suggestion about heat pump/AC unit. (Not applicable in this case.)
Suggestion about booster heater for dishwasher, sump pump, or sewage
pump. (Not applicable in this case.)
Suggestion about someone illegally tapping power to a nearby house.
(Not applicable in this case.)
Suggestion about on-demand hot water. (Not applicable in this case.)
Suggestion to use a signal tracer (such as a Fluke Networks Pro 3000)
to follow the circuit.


Resolution (somewhat): it turned out to be an in-floor heating unit in a bathroom. There are still unexplained variations in power consumption (suggestion to get a clamp meter); perhaps the OP will be able to provide more information if anything comes up.