NEXUM had been tasked, though Kickstarter, to come up with the​ ​AQUA​ ​microscopic battery-less headphone amplifier. Although initial crowdfunding seems to have already finished, there like to be an updated version with improved sound quality.
In terms of battery size, sound quality is by weight, but the button seems like a sensitive trade-off, and may even be a permanent feature if with headphones.