Though not strictly a module, more a pragma, use strict; is still by far my favorite :).

DBI (you should make all modules link to CPAN), is mostly hidden behind some other module in our development environment. So next to strict, in order of favorite-ness, that would be warnings, Getopt::Long, Data::Peek, Text::CSV_XS, DBI, DBD::Pg, DBD::SQLite, DBD::Unify, DBD::CSV, autodie (likely to rise in this order), JSON, WWW::Mechanize, HTML::TreeBuilder, XML::Simple to name just a few.