Joomla 1.6 / 1.7 has a great new feature which allows to expand registration form without any third party plugins or core hacks such as this.

Go to Extensions > Plug-in manager and enable User – Profile plugin
Open the plugin, expand Basic Options, disable not required fields and set required fields to either Optional or Required.

Enabled fields will appear at the bottom of your registration form in User Profile section.

There are 12 commonly used fields available. Unfortunately there is still no easy way to add additional fields (it is possible, just not very straight forward). However you can rename exiting fields to suit your needs. This is done via language override. Read this article to find out how.

Joomla 1.6
Joomla 1.7