Fat is about 50 times as energy dense as the best lithium batteries. Even if the conversion efficiency were only 10%, that's still a good 5 times advantage.
So as it turns out, the obesity epidemic could well be the solution to powering portable electronics. There's already a product out there that does it:
Charge your phone using pedal power with the K-TOR Power Box
The obvious flaw is that while it's reasonably easy to generate 50-100W with a pedal generator, most tablets only accept charge at 10W or less. They also expect a very stable supply so something like a laptop (which can accept as much as 90W) won't charge properly. That's probably why that device is limited to only 20W, in order to account for inrush.
Maybe the solution is to develop something like that but with a built in high current battery. Then it's just a matter of a 30-45 minute workout in order to charge your device for a few hours.
Back when I worked at Intel, their gym had some rather interesting stationary bicycles with USB ports for charging your smartphone. (I don't think it has a built in battery.) So the idea works, we just want it in a more portable form factor.