If it's a true boost converter, it's also going to need an inductor somewhere. If it's really just driving an LED, it could be a charge pump, which uses capacitors instead. But charge pumps probably can't handle the current level that a phone requires.
"If it's a true boost converter, it's also going to need an inductor somewhere."
We use resistors as inductors. Plenty of wound resistors in teeny-tiny packages exist, that's how we have metal detectors with credit card-sized boards, the literal bulk of the unit is the frame and coil assembly and board housing + adjustment controls.
Older gas charge pumps can handle it, but they're bulky and they are lossy and generate a lot of heat.
