Mozilla uses WebTrends for most of its web metrics like visitor and session analysis. We use SiteSpec for multivariate testing.
For everything else we use Kimball's dimensional modeling to build data warehouses and then build reports, dashboards, and analysis off of them. We make extensive Pentaho's suite of tools for ETL, multi-dimensional querying of OLAP cubes, report creation/publishing, and a website to deliver interactive dashboards.
I would strongly recommend that anyone asking these questions do some extended web reading or pick up a few books on the topics of ETL, Data Warehousing, and Business Intelligence. I feel it is critical to have experience these fields if you need to derive insight from data. Using open source ETL/BI tools can help a startup without blowing the budget. When they discover the value for their company, they can look for specialists to join the team or they can investigate support and contracting options available.
In particular, a good ETL tool like Pentaho Data Integration is invaluable to any company that has to do anything with moving data from one place to another. Even if you don't use it for traditional BI.
For our Socorro project, we are making use of a Hadoop/HBase cluster to store large amounts of crash report data. The next two versions of Socorro will be pushing significantly more analysis into MapReduce and Hive type queries.
We are also working on a new data storage and analysis backend for Test Pilot which is based on Riak. The interesting part on the analysis side is that it is performed by submitting MapReduce jobs vis a simple HTTP POST to the REST API.