I ran across this message while googling the same issue as the basic math seems to contradict examples of what projects people have created. I found a video demonstrating 20 fading LEDs being attached to the LilyPad unit. I'll post the link below since I can't embed a hyperlink in my first forum post. 
The LEDs in the video are the "LilyPad" style ones that come with the resistor already on board. It appears that the power supply in use is the Lithium Ion battery, run though the LilyPad LiPo PS unit, which supplies the same 5v as the AAA PS, but for a longer period of time.
The LilyPad can also be run off of 3.7V coin batteries, but I'm not sure how much light you'd get out of 20 LEDs, or for how long.
The project I'm working on involves at least 12 LEDs run off the LilyPad, and needs to have the smallest power supply form factor possible. I'll let you know how it turns out when I prototype the circuit and test the longevity.