How long can I leave my Prius c without it being started or driven?

You have two batteries: the 144V high voltage one, and the 12V auxiliary one. It is the 12V that is susceptible to failure. At this stage, you can comfortably leave it for two weeks. Longer than that, you need to look into a charge maintainer, like Battery Minder, Battery Tender, etc.

 

The 12V battery is small in capacity compared to a standard car.

The quiescent current draw is high, especially in 2004-2009 Prius. Toyota has done better recently with the newest models but having the 12V in the passenger compartment (under the rear seat?) means that it had to be fitted with a small 12V AGM battery. The charging current is also very small, such that a vehicle that is seldom driven will have its 12V die, because unlike cars with an alternator that pumps out 50 amps when first started, the DC converter will only charge it with 3.5 amps.

Toyota recommends disconnecting the 12V negative if the car is being stored for 30 days. I think that is overly optimistic, to expect the 12V to last that long.

I should mention, the HV battery can go months and not be harmed. Toyota recommends starting the car every 60 days, which is conservative.

 

My 2009 prius will charge its fully discharged 12 battery at overr 40 amps...
This is seen watching scanguage hv current go to 4.5 amps. 3.6 amps over normal prius hv current of .9 amps..
3.6 amps at 230 volts (approx hv fully charged voltage) = 828 watt of dc converter power all due to discharged 12 batterry...
Factor in a 70 % effecency factor for dc to dc comverter this leaves 579 watts of charge to 12 volt battery.

579 watts divided by 14 volts = 41.4 amps.
The dc/dc converter is rated at 90-100 amps fused by 100-125 amp fuse.

You stated prius only charges at 3.5 amp rate is incorrect.
While a fully charged battery will trickle at about 3.5 amps a dead will pull over 50 amps.
While toyota recommends the 12 volt battey be bench charged at 3.5 amp rate...
The gen II will charge a fully discharged battery in about 90 minutes with initial charge at 41amps quickly falls to 30 amps after 5 minutes and 25 amps after 10 minutes the charge current diminishes to around 10 amps after the first hour..and becomes fully charged at approx 1.5 hours...

I would think all prius models have similar dc/dc converter ratings as the gen II..

 

Also you can shut of the Smart Key System while you are away to reduce its parasitic load.