Soft 404
Nice. I like the lamp shade, nice touch. Just needs the pull cord switch 
It was suggested here that you may be able to get down to 200mV:
May be able to fit that toroidal core in the base ?
Excellent reference! Karen did a wonderful job on her joule-thief circuit. I thought about mentioning it, takes for mentioning Karen's project!
These modules are beginner friendly and provide a regulated stable DC output. So you could also run a neopixel and small mcu off a single alkaline cell.
Yes, past 600 mV there isn't much left... Of course you can use a fresh cell too
Some of the low cost dollar store solar lamps use tabbed ni-mh cells, I'm sure if I were to shop around I could find them for pennies...
