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It is possible to construct a very large electrolytic cap quite easily.

Materials:
Roll of  paper - I use brown kraft paper, must withhold wet fluid. I soak paper in electrolyte for about 1 hr to get good saturation.
Roll of Mylar plastic
Electrolyte - 50% Antifreeze
ribbon wire
12 ga insulated wire for post.
Empty can with lid (paint can Hardware Store, remove inside paint with wire brush or paint strip acid, recycle canned food, aluminum soda cans etc. - best result with aluminum) .2" taller than paper.

Roll paper and mylar creating a spiral coil - bi-filar, Mylar must be shorter than paper as to avoid shorting if it contacts the top of the metal enclosure. Make sure that the plastic side face paper and the aluminum layer is inside. Keep the coil as tight as possible and make enough to fill the can leaving .2" space when placed inside the can.

Attach ribbon wire with cellophane tape to each side of the mylar film, ensuring that the wire cuts down at 90 degrees perpendicular to the coil. Attach ribbon when coil diameter reaches hole for conduction post, just like in a electrolytic cap. Drill 2 holes on can cap, hole size to tightly fit conduction wire. Finish wrapping the coil. Wrap the outside of the coil with cellophane tape securely and place in can. Solder ribbon to conduction post. Fill with electrolyte and cap the bottom end to prevent leakage.

I am testing electrolytes to see which one works best. Mylar is cellophane dielectric with aluminum coating. There are many peculiar electrochemical behaviors of Al such as self induction etc, I am testing acidic and alkaline electrolytes. Anyone got a link to more info will be greatly appreciated.

For charging 12v Lead Acid, you need to construct 1 gallon size caps, wire in series for faster charge time.
Pesto, Ultra captret

hi hidave,  thanks for sharing your setup.  Can you say what exact diode number you use? 
   I also just have one suggestion that would further prove what you described.  Get two batteries at the same starting voltage.  Run both down to the same amount.  Put one in your circuit and let the other just sit.  Measure voltage on both after the one in the circuit is charged back up.  Most alkalines will recover a fair amount of voltage after sitting a while even though they are not quite as strong in providing current.  I think this needs to be ruled out before getting too excited about this.  I'll try getting time to do something like this. 
   Also thanks for the reminder on Cap building as I recently found a big roll of metalic mylar stuff at a garage sale.  I know it's a huge cap just waiting to be built

broli,  I don't think you have polarities correct from what hidave said.  Thanks for the diagram but check the polarities.

You are correct. However this setup makes the led shine brightly too. It's crucial to connect the plus to the housing or else it won't light, or barely that is.

This picture might help.

A bank of 4 x 250v 2200 uf caps wired in series will charge a dead 9v cell from 5v to 9.5v in 2-3 hrs. I've tried!.

I assume you confirmed that this is true usable charge that can give as much (or more) joule energy as when it was conventionally charged?

I assume you confirmed that this is true usable charge that can give as much (or more) joule energy as when it was conventionally charged?

currently draining a charged battery with LED. Will let rest 24 hrs and recharge as requested by  ibpointless2

battery is very cold to the touch, these are regular alkaline batteries.

I'll do a controlled test later and post the joules data, or perhaps someone can do it?

I just did a quick and dirty test on a 9 volt battery that has been sitting a couple days.  Start was 9.09 volts.  Hooked up a 330 uf electrolytic same as my diagram - no diode.  Voltage went down to 9.08 volts.  Didn't see any rise after a few minutes.  No time to check further right now but as I was saying I'm concerned about alkaline self recovery in this situation.  My quick test is not conclusive at all but just hints that it might be involved.

small caps takes long long time to notice gain, get a hold of 200v 2200 uf caps, PC power supply have some decent size caps. Or make your own. I have tried a 12 oz red bull can captret that I made, works very well.