You can use something like the Chrome Developer Tools to view, pretty print, debug, or step through the JavaScript being used by Salesforce to make it easier to comprehend.
It looks like the JavaScript responsible for popping up the activity reminder window is in the sfdc/main.js file in the code related to the ActivityReminder
.
You can execute the following JavaScript from the Chrome Developer Console to get the Activity Reminder Window to pop up.
ActivityReminder._showPopup(true);
It will just display sample test data:
Also, the Chrome Developer Tools can give you a good view of the JavaScript that is written directly in the page (as opposed to included from .js file) and a view of the actual HTML of the page.
On the account detail page (standard layout) in my DE the ActivityReminder gets initialized in a function called bodyOnLoad() which is called from the body's onload() event.
Although it could be useful to understand what's going on behind the scenes to diagnose your problem, you should not write any code that depends on / uses any of the built in JavaScript. It is not a published Salesforce API and, therefore, could change at any time. You could end up in a situation like chatterfeed toggle not working post Spring '13.