<p>What would be the best way to develop a text box that remembers the last x number of entries that were put into it.  This is a standalone app written with C#.</p>

This is actually fairly easy, especially in terms of showing the "AutoComplete" part of it.  In terms of remembering the last x number of entries, you are just going to have to decide on a particular event (or events) that you consider as an entry being completed and write that entry off to a list... an AutoCompleteStringCollection to be precise.
The TextBox class has the 3 following properties that you will need:
AutoCompleteCustomSource
AutoCompleteMode
AutoCompleteSource
Set AutoCompleteMode to SuggestAppend and AutoCompleteSource to CustomSource.
Then at runtime, every time a new entry is made, use the Add() method of AutoCompleteStringCollection to add that entry to the list (and pop off any old ones if you want).  You can actually do this operation directly on the AutoCompleteCustomSource property of the TextBox as long as you've already initialized it.
Now, every time you type in the TextBox it will suggest previous entries :)
See this article for a more complete example:
http://www.c-sharpcorner.com/UploadFile/mahesh/AutoCompletion02012006113508AM/AutoCompletion.aspx
AutoComplete also has some built in features like FileSystem and URLs (though it only does stuff that was typed into IE...)