While it's not possible to add a Visualforce page or component to the standard Salesforce home page, it is possible to inject straight HTML and JavaScript into the home page via an HTML Area component. Furthermore, you can use Salesforce's AJAX Toolkit to connect to the API, which opens up a world of dynamic possibilities for your Home tab.
There's only one trick to connecting to the API: You need the current user's session ID. Luckily, this information is stored in the sid cookie and can be retrieved and easily stored in the sforce.connection.sessionId property.
From here, you can use all of the tools in the AJAX Toolkit to bring your data to life on the Home tab.
