Below are the following open source scripts that we are familar with modifying and using.

Joomla! is one of the most powerful Open Source Content Management Systems on the planet. It is used all over the world for everything from simple websites to complex corporate applications. Joomla! is easy to install, simple to manage, and reliable.
Homepage: http://www.joomla.org/

WordPress is a personal publishing tool with focus on aesthetics and featuring cross-blog tool, password protected posts, importing, typographical niceties, multiple authors, bookmarklets.
Homepage: http://wordpress.org/

Crafty Syntax Live Help - A Live Help chat system featuring monitor your visitors, proactively open a chat session, multiple chat sessions, referer tracking, page view tracking, multiple operators, canned responses/images/URLs, multiple departments each with different icons, leave a message.
Homepage: http://www.craftysyntax.com/

phpBB2 - A widely-popular open-source bulletin-board package, works well, simple user interface and admin panel, clean look, scales well, and can be customized.
Homepage: http://www.phpbb.com/

OS Commerce - A power-user shopping cart with a big variety of modules and support of almost every payment gateway. A big developers community is ready to offer custom solutions depending on your needs.
Homepage: http://oscommerce.com/

dotProject - Project Management featuring companies, projects, tasks (with Gantt charts), forums, files, calendar, contacts, tickets/helpdesk, language support, user/module permissions, themes.
Homepage: http://www.dotproject.net/

TikiWiki is designed to be an international, clean and extensible Content Management System and Groupware that can be used to create all sorts of web applications, sites, portals, intranets and extranets. TikiWiki also works great as a web-based collaboration tool. TikiWiki has a lot of native options and sections that you can enable/disable as you need them.
Homepage: http://www.tikiwiki.org/

Open-Realty - Real Estate listing system featuring attachments, flexible search, template system, Yahoo Maps interface.
Homepage: http://www.open-realty.org/

PHPauction - The GPL version of PHPauction featuring email notification of bids, attachments, reserve price, minimum bids, standard and dutch auctions, bid history, send auction to a friend, highest bids, auctions ending soon.
Homepage: http://www.phpauction.org/

Moodle - A course management system designed to help educators create quality online courses. Available in 34 languages and features a WYSIWYG HTML editor. The teacher has full control over all settings for a course. There is a flexible array of course activities (Forums, Journals, Quizzes, Resources, Choices, Surveys, Assignments, Chats, Workshops), user logging and tracking, mail integration and much more.
Homepage: http://moodle.org/