So @EricH gave me an interesting idea.  He asked if using an intermediate step with another set of capacitors could help with the energy transfer problem.  If you could find large-valued capacitors with low self-discharge, you could take a long time to charge those with a coin cell, then charge the supercapacitor quickly from the intermediate caps. It sounds like it could work, but I don't think the right capacitor exists for this intermediate step.

What about using a rechargeable battery as the intermediate energy storage?  This gets interesting.  Everyone seems willing to allow an electrochemical capacitor as intermediate storage, so why not a rechargeable battery? (I'll refer to the coin cell as a "cell" and the rechargeable battery as a "battery" in the discussion below).

Let's forget about any technical problems for a minute and consider the contest judges and spectators.  You have to convince them somehow that you're not running anything from energy pre-stored in the battery.  Since state-of-charge is very difficult to measure accurately, I'm not even sure I wouldn't be cheating with most battery chemistries.  The exception is NiCd, which can and should be stored with the terminals shorted and at a zero state of charge.  It's how NASA stores their NiCd cells, as detailed in this technical report on NiCds.  So, if I take a couple of AA NiCd's that have had their terminals shorted for a few days, then verify there is 0V across them, I think I can make a convincing argument that there's no energy hidden up my sleeve.

OK, so there's a way to verify that all the energy is coming from the coin cell.  What are the properties of a NiCd battery? 

• Nominal voltage 1.2V with a flat discharge curve.
• Can sustain very high rates of discharge (think of a cheap cordless drill).
• Self discharge rates quoted as 10% per month (wikipedia) or 1% per day (NASA TR).
• Tolerant of varied charging methods (C-rate and end-of-charge detection)

Overall, they sound like a good intermediate reservoir for energy storage.  They have a much lower self-discharge rate than supercapacitors, so can be charged slowly from a coin cell without terrible losses (a DC-DC converter is still required).  Then, once charged, they can be drained very quickly to charge the supercapacitor before supercap self-discharge becomes an issue.

What are the drawbacks?  First, the energy will be going through two DC-DC converters, so losses get compounded there.  Also, there's the charging efficiency of the NiCd's. Wikipedia mentions that at a C/10 charge rate, you have to apply around 1.5C of charge to fully charge a NiCd (equivalent to a 33% loss of energy).  The NASA TR, however, shows that this ratio is a strong function of temperature (P. 13).  With battery temperature near 0C, the ratio approaches 1, so much less energy is lost in charging.

So, can I take a 1.7Ah LiSOCl2 cell, charge some 1000mAh AA NiCd's, then use the NiCd's to charge a 67F capacitor to 14V?  Here's how everything stacks up:

Storage	Capacity (mAh)	Energy (J)
CR2477	1000	9000
TL-5935/P	1700	20000
2x NiCd AA	1000	8600
3x NiCd AA	1000	12960
4x NiCd AA	1000	17280
67F capacitor @14V	-	6566

From the TL-5935/P datasheet, it looks like I can get the full 1.7Ah from the cell at 10mA, which is the maximum recommended continuous drain.  I also estimate that the cell voltage will remain stable at around 3V for the entire discharge.  Discharging at this rate will take 7.08 days.  Assuming the 1% per day self-discharge rate for NiCd's, I might lose 600J during this week. Assuming a 33% loss due to NiCd charging inefficiency (which might be improved by cooling), and a 70% DC-DC converter efficiency, I end up with 8961 J in the NiCds, which almost fits in 2 AA's. I'll call it 8600 which just fits.

In order to charge the capacitor to 14V with the 8600J in the NiCd battery, I need a DC-DC converter with around 76% efficiency.  Since the NiCd's can tolerate high current drain, I can use an efficient off-the-shelf converter, so that might not be too difficult to achieve.

It ain't pretty, and there's not much slack to play with, but it looks possible to start a car with a coin cell again.

EDIT: I found that charging efficiency for NiCd cells is nearly 100% until they reach 70% capacity.  So, I can gain a lot of efficiency by using more NiCd cells and charging them to less than 70% capacity.  Nice.

Also of interest is the PowerSonic NiCd Catalog.  I need to pour over those graphs.