Piwik can. Run a self-hosted instance or pay them to run it, and it handles server logs no problem.
You do give up the ability to get live stats, but you get better performance and the ability to track more visitors (read: people like me who have blocked GA for years...)
Added bonus: Referrer spam is automatically blocked by default.
EDIT: Almost forgot GoAccess, if you are okay with a terminal app and want live stats (can also be scripted to generate HTML reports) - http://goaccess.io/
Piwik is astonishingly good. It'll give you trouble if you have a lot of traffic, but for moderate sites it's pretty much a completely self hosted drop in replacement for Google Analytics. You can track conversions, set up campaign goals and view statistics overlays for example. Not all features are available when using the log import backend, as compared to the javascript thingy, but most are and it'll give you an overview on how your ad blocking customers are doing.
Quick note that with hourly, or more frequent, cron jobs you get quasi live data. Mentioned a little about logimport in a recent post on stripping external calls and pointless js/css/font loads from our site. Otherwise well put, same experience here.
Btw, tail -f /log.log is always fun for live data...
Yeah, I build a system in the late 1990's that did something like that. Required a dedicated server though, the web server farm would copy over their log files to this analytics machine with cron jobs, which would then dynamically aggregate that data and update the stats cache.
Charles, Snipcart co-founder here. As mentioned by the end of the post, we're actively looking to expand our analytics game a bit. Piwik seems like a good place to start, so thanks for sharing. Cheers!
Try running it alongside GA and comparing stats. I did this for a while and switched to Piwik as it consistently counted more visits (legitimate visits, obviously).
Also, please consider adding your Google Analytics use to your privacy policy. They require it, even though most people ignore this requirement and there is no enforcement.
Side note, I just tried @ replying to your companies post on Twitter and got a @your account may not be allowed to perform this action' http://i.imgur.com/9uoEn9B.png Strange, never saw that before...
We received some mentions in the last minutes, and we just reviewed our account settings, nothing seems problematic on our end. Weird indeed. But thanks for the feedback!
That's an important point; I'm seeing referral spam in my GA reports which I never noticed before 2015. I manually discount it when compiling reports, and my understanding is that these spammers hit GA UAs at random without even loading your website.
The same way you can gather all available stats on server side and push these yourself. You can override IP too. Obviously not everything will be available, like screen resolution, but you still can capture a lot of stuff.