I've been running just a super capacitor in my motorcycle for 6 months and am very pleased with the results. Ditching the lead acid battery, I was able to save 10 pounds, and the super capacitor should outlive the bike. This success encouraged me to experiment with car battery replacement.
This video shows how I have setup a super capacitor and small lithium iron phosphate battery to start my car. To protect the LiFePO4 battery, I have installed a 0.2 ohm 50 watt resistor in series between the super capacitor and battery. This limits the current going to the battery, which should greatly prolong the life.
I measure the peak amp draw from starting the car, along with the peak charge/discharge on the LiFePO4 battery.
The TSX has a parasitic load (car turned off) of 0.042 amps. Therefore, the 4.2 amp hour LiFePO4 battery can only supply energy for 100 hours, or about 4 days before going dead. That means I need to charge the battery if I intend to let the car sit for more than 4 days. I could also add a larger battery pack to extend the amount of time the car can sit.