[scrumper]

There isn't a hope. The MP3 player itself, with solar panels, encased in glass? It could last a good long while... if it weren't that it'd be buried by sand very soon.

The mechanical transducers in the loudspeakers are the big weak point though. Suspensions, foam mountings, the voice coil magnets themselves, the cone; all subject to stress. Under ideal conditions, hundreds of thousands of hours of use. A desert is far from ideal - fine dust and temperature extremes - though far from disastrous too (a damp salty cave by a tropical ocean). Hundreds of thousands of hours gets you a decade and a half of play. So only 54,999,985 years to engineer for.

Alternative speaker tech (piezo speakers maybe?) could last longer. But it's limited! Fine moving parts are anathema to this kind of longevity experiment.

EDIT: Actually I'd encase the electronics in a giant diamond; glass would get scoured rough by the sand pretty quickly, to the detriment of the PVs.

[hangonhn]

But why not go nuclear? Not a full on nuclear reactor but one of those nuclear RTG batteries. The half life might be low but could you not keep doubling the amount until you reach the requisite years? Basically take the amount of Pu you need to power the device, and double it 20 times (half life of Pu-238 (87 years) * 2 ^ 20 >= 55 million years). This way the entire unit is self contained and you can encase it in whatever super durable material you can think of -- artificial diamond/sapphire.

[yellowapple]

Hell, ain't Africa home to some giant natural fission reactor? Reignite that thing and there's your power source.

[scrumper]

Because the power source isn't the problem. It's the mechanicals in the speaker.