High voltage input and low voltage capacitor in series safety
My friend built a very similar concept for his boat and it worked fine.  So what you are attempting may be possible.

The first thing to note is that you only want to power your stereo.  You don't want your capacitor to try and power the starter.  You can isolate your capacitor and stereo from the main 12V via a diode.  That way your stereo and capacitor can keep their voltage high when the input voltage from the battery dips.  If you don't want to use a diode its your choice, Maxwell makes some nice automotive starting capacitors (300F range) but they are several thousand dollars.  Your radio by itself probably needs just a few Farads.

The next thing to do is find out how much capacitance you actually need.  It may be that the amount of capacitance is way too big to be practical so don't even bother.  On the other hand it could be that you don't need several farads and you may be able to buy a single 1F cap rated to 25V (so no need to worry about your stacking problem).

To determine capacitance you need to know three things.


How much wattage the stereo is drawing.

a)  Measure the battery voltage (V_batt) powering the unit.

b)  Insert an amp meter in series with the 12V power leads on the
stereo.

c)  Measure the current draw with the unit on.

d)  Power = Current x V_batt
What voltage does the stereo actually reset at?

a)  Find some way to gradually reduce the input voltage into the
stereo and see at which point it resets (V_reset).


3)  How long you crank for (t_crank)

After you have V_batt and Current and V_reset you can calculate capacitance.  I am going to make the assumption that the stereo draws the most wattage at 12V and lower wattage at lower voltages.  In the worst case you then have...

C = 2 * t_crank * Power /(V_batt^2 - V_reset^2)

So for example if the battery is 12V and the radio resets at 8V and the radio draws 50W and you crank for 3s then the capacitance becomes...

C = 2 * 3s * 50W / (12V ^2  - 8V ^2) = 3.75F.

You would want to pick the capacitor to be larger so that you have some margin (say 5F in this case).

Note that you may need to make a small correction to the formula above to account for the diode drop (depending on which one you use).  Also you may want to use two diodes and a resistor to limit inrush when you first connect your capacitor.

If it turns out that you need several farads and you need to stack them, then it can be safely done.  What you need to do is make sure the voltage across them remains balanced.  This can be done by putting a Zener diode in parallel with each capacitor.  The zener diode should have a breakdown voltage less than your capacitor rating.  Also the sum of the zener voltages should be greater than your battery voltage.

The capacitors and zeners shouldn't draw any significant power when the car is off.