Is there such thing as a freewheel that doesn't click?
As already said, Shimano used to manufacture Silient Clutch rear hub. But that has been stopped a few years ago, so if you manage to source one - you are lucky man. 



I used to have one of them and it was truly silent. It was heavier than standard shimano LX hub, but it was silent and with instant engagement.

Also I used Chris King hubs. If you put a lot of grease in their mechanism - they also become silent. 
Pretty much with most of the freewheels, if you put a lot of thick grease in them, you'll silent them for some time. Unless you ride Hope hubs -)

I've heard about Stealth hubs - they are claimed to be silent, but I never had a chance to try them out. 

You might as well read this thread about silent hubs for further information.