Codeigniter for large projects!

I use CI for a national site that meets your criteria for a "large project". Works great! Very fast. Tens of millions of page views/month. We haven't upgraded to CI3 yet as it was just released and may not be able to for some time due to the large codebase and number of changes required for the v2 -> v3 upgrade, so I'm speaking about the v2.x branch. As far as the large/complex queries, that really doesn't have to do specifically with CI but is more up to how optimized and properly indexed your database and tables are as well as the hardware you're running it on. But yes, some of our tables have tens of thousands of rows and have 6 or more joins in queries, doing complex statistical calculations, with a large amount of data. This is a multi-million dollar generating site.

Of course some of the larger, more complex queries don't use active record, but most are using active record (query builder in CI3+)