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/
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.
https://piwik.org/log-analytics/
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/