Actually, we already have such things like alt.anonymous and they work well in practice; see my just submitted https://news.ycombinator.com/item?id=6819799

Except you'd have to spend some money, so the origin would always be known.

There are still some Bitcoin Faucets open, such as FreeCoins[1]. They give out almost nothing, but it's enough to make transactions just for the registration, and you don't need to provide any information.

[1] http://freecoins.herokuapp.com/