AWStats (http://awstats.sourceforge.net/) is decent if you just want some simple log-based analytics.

Don't know what you mean by limited, but logfile-based analytics software doesn't have access to as much data as software that can use other data sources like cookies (ie it has difficulty tracking sessions).

If I might ask, why exactly do you need to make your analytics completely undetectable to users? In most cases tracking pixels and the like have no effect whatsoever on the user experience.