What are some examples of filter Correct faulty and chopped off HTML?
In Drupal, there is rule that a text filter should never change anything that the user saves.  The "Correct faulty and chopped off HTML" follows this rule. That is why you don't see any changes in the body field when you edit.

However, the filter will fix your HTML when the text in the body field is rendered (i.e. displayed to the user). 

For instance, if you use the Drupal 6 or 7 "Filtered HTML" text type (which makes use this filter) and put the following faulty and chopped off HTML in the body (this is all you put in the body of a node):

&lt;ul&gt;&lt;li&gt;xyzzy


and examine the rendered HTML in your browser (for instance using Ctrl-U or Firebug in Firefox), you'll see that it is corrected to:

&lt;ul&gt;&lt;li&gt;xyzzy
&lt;/li&gt;&lt;/ul&gt;


Screen shots

Text in editor:


  


Rendered text, as displayed by Firebug: