Since the auxiliary battery's duties are minimal, it is relatively small by automotive standards (~30 Amp hours). This isn't a problem for most drivers, but I travel a lot and don't drive much even when I'm around. Consequently, it's common for the battery to discharge below the point where it is capable of starting the car.
Discharge is due to powering the vehicle alarm, clock, radio settings backup etc. when the car is turned off. The aggregate demand of these services is not negligible. I haven't measured it, but others have and found a figure somewhere around 20 mA. It doesn't seem like like much, but it adds up:
30 Ahr / 0.02 A = 1500 hours = ~62 days
62 days is a reasonably long period, but keep in mind that the battery may not be fully charged to begin with and it will fail to start the car well before it is 100% discharged. A more realistic estimate of a safe non-drive period is around three to four weeks.
Unfortunately not driving my car for that period is a regular occurrence and I am thus often presented with a flat battery. I killed the stock battery that came with the car due to repeated deep discharge (and high current charging the depleted battery after jump starting).
I replaced it with a deep-cycle (more resilient to deep discharge) yellow-top Optima. Although more tolerant of deep discharge, it's still not a good idea to repeatedly fully discharge it. Moreover charging the deeply discharged battery at high current levels (as the car will do once started) further damages it.
See here for all you ever wanted to know about lead-acid batteries.
Starting a Prius with sub-C cell batteries
My usual approach for dealing with a flat battery is to start the car using a battery pack that is intended for radio-controlled cars.
Robin Mitchell helpfully went to the trouble of measuring the Prius startup current. Based on the results, a typical startup sequence draws 15-30A for several seconds (although occasional brief spikes to 60A were seen).
This is a trivial amount of energy, but the magnitude of the current is beyond many small batteries, which will exhibit too great a voltage drop. High quality sub-C NiMH cells can have internal resistance as low as 2-3 milliohms, providing sufficient current for the startup requirements. A pack of 10 cells (as pictured above) provides a 12V battery that can be used to start the car.
Worst-case voltage drop = 10 * internal resistance * current = 10 * 3 mΩ * 60 A = 1.8 V
Although this works in a jam, it's not a long-term solution as the auxiliarry battery will be damaged by the repeated deep discharge/recharge cycles.
Harvesting the sun
A float charger connected to mains would be ideal to keep the battery topped but, but my car is parked on the street with now power access. The next best thing is a solar-powered charger built around the Analog Devices LT3652.
The datasheet does a good job of detailing reference circuits for the majority of applications, so there is very little in the way of design to be done, aside from routing a suitable PCB:
After some stove top reflowing (I had yet to build my toaster reflow oven), I had a compact solar charging module that - in combination with an 18W solar panel purchased on AliExpress - would keep the auxilliary battery safely topped up.