After searching the net for quite a while I could not find a pure wireless music player for small money. In case you want to equip multiple rooms in your house with a music player it will cost you a lot of money. All products on the market come with a lot of features, of which most are not necessary. The following links show interesting hacks on how to build your own WiFi music player based on OpenWRT:

Using routers as a platform is a very good choice as they produced in high volume and thus the price for the
hardware can be very low. Every platform evaluation board is much more expensive, especially when you
want to have WiFI connectivity (> EURO 150). Those evaluation boards or development kits are produced
in very small volume and thus the price is not competitive.

I decided to use a router as the base for my player. My current setup is this:

This makes EUR 66,56 for the music player without loud-speakers. I guess the price can be further reduced
because the sound-card was more expensive than expected (additional shipping costs), and there are much
cheaper USB Hubs available, but this one I already had around.
The router TP-Link TL-WR1043ND is the TP-Link flagship consumer product, choosing another (maybe a
used one) one where you can easily solder a USB port, would also do.

For the second version of the music player I expect the price to drop around EUR 30.