How do I eliminate previous searches from showing up in the search
window when I enter the first letter (or two) of a search request?

smithden,

Before I begin with my advices, please let me clarify, that Google
Answers Researchers are not Google Employees. Researchers have vast
knowledge on this and other issues (and have been carefully selected
by Google), but do not represent Google nor do they have any "inside
knowledge" about Google's policies.

With that aside, this is actually one of the top 3 FAQ (Frequently
Asked Questions) asked to Google.



This is actually a feature of your browser, not of Google. 
To Fix on a Windows PC:

Open Internet Explorer and go to the Tools menuat the top of the window
select Internet Options from the drop down menu
select the Contents tab
Under Personal, select AutoComplete 
Click on Clear Forms
You can also uncheck the Forms box in this same window to keep this
from occurring in the future.
://www.google.com/contact/search.html#1


If this answer requires further explanation, please request
clarification before rating it, and I'll be happy to look into this
further.
Nenna-GA
Google Answers Researcher

Just use your arrow keys to select one, and then hit DEL on your keyboard.

If you want to disable autocomplete, in internet explorer, go to tools
> internet options.  Go to the content tab, and click autocomplete. 
Uncheck any options you dont want (you can also click the clear forms,
and clear passwords buttons)