The cheapest amp you'll find is a PAM8403:
http://www.ebay.com/itm/171853624163
less than a dollar on ebay.
You can connect it directly to analog out of the RasPi - for the cheapest possible solution (solder wires between the RPi analog out - on solder points below the analog jack on the RPi - and the iput of the amp).
I have several of those, and can honestly say that they are quite OK - and those 3W sound a lot louder than you'd expect. Another plus is that this amp is powered by 5V - so you can use the same power source as for the RasPi.
Can't go cheaper than that... And those amps are also tiny, so they will fit in the same enclosure with the RPi.
BUT - you'd need a relatively solid power supply for the RasPi - 2A minimum.... since it will also power the amp.
If you want a better sound quality, go for the cheapest DAC you can find - the cheapest I2S you can use is Sabre ES9023 - check it out on ebay, about $14.... Wire it directly to RPi GPIO pins, and solder PAM amp input to RCA audio out on the DAC, instead of to the RPi analog out.
