Can anyone tell me what is the CHEAPEST way to play loud recorded music in a large garden without having to resort to expensive amps and PA systems?

Rent a system for the event.

If this is a short term problem, like a party or wedding, I recommend finding a rent to own store in your area.

Most of them will rent a big stereo system for a week or weekend.

If you need a permanent solution, an expensive amp or pa system are the only options I know of.

Are you more concerned that the music will be heard or the quality of the sound?

More concerned it will be heard!