Well, we have daily update requests (it's not a web app), so we knew the number of daily users already. But we really wanted to know the monthly users, gives a way better idea of the actual user numbers. What we ended up doing is to transmit the timestamp of the last update check with each update check, that way we can see which requests are the first over an arbitrary period.

Granted, the situation is probably not that similar to what DDG faces, but it was just one issue that came up. What we use for web analytics is based on awstats, which operates on access logs with anonymised IPs. Not as good as Google Analytics, but as useful as it gets without JS/tracking, I guess.