Many setup scripts for SQL Server include the 3659 trace flag, but I could not find official documentation that explains exactly what this flag means.
After a lot of research, I found a reference to this flag in a script called AddSelfToSqlSysadmin, written by Ward Beattie, a developer  in the SQL Server product group at Microsoft.

The script contains a line which suggests that this flag enables logging all errors to errorlog during server startup. I’m unsure of what kind of errors are not logged without setting the flag, but I didn’t find any further reference. The BOL page for Trace Flags doesn’t list this one, so if you happen to know something more, feel free to add a comment here.