Here is a web page that describes a 0.5 volt to 5-6volt booster circuit.

http://www.discovercircuits.com/H-Corner/05-6-con.htm

Quote from page.

It turns out that TI's 74AUC family of parts can work down to about 0.45 volts. I tried one of their single schmitt trigger parts and found I was able to make on oscillator function nicely at 0.5 volts.

I was thinking of trying a CD4049 to see what it could operate at, I have some 74 parts but not many,

..

Yeah that part is the one to get and should be cheap 57 cents each or 40 cents each if you get 10.  With one of those parts we
should be able to make an oscillator similar to the one below but run it from less than 1 volt.

I've made an efficient oscillator circuit from a CD4049 chip and it works ok at 2.5 volts but won't work much lower.

After adjusting it a bit I got about 90% efficiency lighting 2 x 5 mm LED's from the 2.59 volt supply. I knocked off 0.05 volts from
the output so it didn't look so efficient, hahahahaha see the shot says 8 v.

The input was 3.3 mA at 2.59 v = 0.00855 W. Input fairly smooth. Measurements done with an accurate DMM and current sensed
across a 1 Ohm resistor.

The output was 1 mA at 7.95 v =  0.00795 W.  Output was DC in a cap as the shot shows.

0.00795 w / 0.00855 w = 93 % efficient. LED's were bright enough to leave dots on my eyes. 

At other less than ideal adjustments I got consistently 60 to 70 % efficient.

First shot shows the collector waveform in yellow and the capacitor voltage that the LED's are running from in blue,
8v across 2 x 5 mm LED's.. I went for zero volts at the collector switching in the first shot.

Second shot is the oscillator signal at the capacitor Blue trace, (second gate of three) and the Base drive in Yellow.

Third shot show the frequency backed off and the coil ringing.

Fourth is the circuit running, the capacitance is 200 pF and I set up the adjustable circuit from the CMOS Cookbook,
the right pot determines the output low time and the left pot determines the output high time, this can be reversed
by using another inverter gate.

The high time pulse output can go to as narrow as 1 uS to over 90% duty and the frequency from 25 kHz up to 180 kHz as it is.

I used only three gates on the bottom side of the HEX buffer.

..

Note that in "CMOS osc 3" you have a very clear ringing that has a frequency very near 400 kHz. This is the natural resonant frequency. At present this is a loss, as power  sloshes back and forth dissipating a little on each cycle and the amplitude of the ringing diminishes to zero before the next "push" happens. 

"Ringy" is good because it means high Q which means small losses per cycle in the resonant ringing. On the other hand eliminating ringing by shortening the time between stimulating pulses is good because then you are approaching true resonant pumping, where a tiny push at the exact right time keeps a high-amplitude response going and this is not allowed to decay.
I think. Maybe.

 
A 74x14 hysteresis inverter version exists too.
 
http://www.digikey.com/product-detail/en/SN74AUC1G14DBVR/296-12880-6-ND/1634643

One would have to experiment to see if the AUC14 works better at below spec supply voltage than the AUC04 .  The hysteresis of the '14 will probably make it harder to start oscillating.

A certain amount of reverse bias across the Base-Emitter
junction is beneficial as it aids transistor turn-off and
enhances switching efficiency.

If the reverse voltage is too great (ordinarily more than
about 8 Volts) the junction will go into breakdown because
its doping is very similar to a Zener Diode.  If breakdown
occurs at the reverse voltage peaks then undesired and
unpredictable things may happen.

Ok Fixed that problem, clamped it at less than 1 volt, and I'm thinking of adding a small inductor (see drawing) I found with a enormous inductance for it's size and turns, must be special core material.

Anyway if I add this inductor at the spot indicated as (A) on the circuit it might help a bit more.

The transistor isn't wired backwards it's 1) emitter, 2) base, 3) collector. all good.  with pnp the emitter goes to most positive, is that right ? Could be confusing.
..

At the moment to get the initial boost to over 5 volts in C3 for the logic circuit I have to stop the pulses from the CD4049 chip by manually holding one of the gate inputs low momentarily, only takes a touch. Needs a fix.
.
Base is blue trace collector is yellow.
..

Yes Mark that coil won't work in a JT for some reason, the resistance is not high, it's fairly thick wire, might need to wind one, maybe a resistor will do. I don't have any Schottkys yet, some ordered but I might get more, this is fun and useful. I need to order the small gate chips so I'll get some of those Schottkys too.

Tinsel I think when I adjust the circuits so that the switch turns on just before the coil finishes discharging then the current is already flowing in the right direction and it gives that push along with it. Even the JT wave form shows that now. You can see that the slope on the decline of the discharge gets clipped by the switch turning back on.

Is that good or better when voltage is zero ?

..

Oh I see, is that transistor bad maybe ? I thought it was happening because of the extra turns on the base trigger coil, the yellow turns are extra ones two blue coils the same on there with the yellow added to the end of one for the base trigger.

..

Tinsel with the Logic circuit I can cramp up the wave form by adjusting the frequency/pulse width and clip the back of the discharge off square, which increases the amplitude of the output, and makes it "square" or rectangle.

..
P.S.Here's a pic of a few of my small inductors, the little one beside the one with blue tape is about a half inch diameter and
has 1.2 mH inductance but only a few turns and hardly any resistance.
Only 7 turns or so and it is really light, like it's core is not made of normal "metallic like" material.
...

The shot below shows the current through the inductor sensed by a 1 Ohm resistor in the place I marked (A) on the drawing. 1.2 mH.
..