In Google, if I type in a letter “a” and nothing else, there appears on the screen a list of all the searches I have ever done that begin with “a”.
How do I clean that up? I tried deleting all cookies but that didn’t work.
This is not a Google feature but a browser/OS feature. What browser, which version, what OS, which version of the OS are you using?
For whatever reason, this computer wont save the search results so I can’t play around with the feature, but have you tried to clear HISTORY yet?
If you’re talking about the Google Search Bar, click on “Google” on the left side. That brings up a drop-down list which includes “Clear Search History”.
That’s never worked for me. Even after clicking clear search history, the drop down list of recent serches still appears.
I had to manually delete all of the past searches. The newest version of the google toolbar gives an option to disable saving past searches.
Go to OPTIONs on the Google toobar, click SEARCH, make sure “Save the search history across browser sessions” is unchecked.
Not that I have anything to hide. 
At the top of your Browser (if using IE), Tools>Internet Options>Content>Personal Insformation>AutoComplete>Clear Forms
If using Apple’s Safari browser, select “Reset Safari…” in the “Safari” pull-down menu.
In Firefox, you just bring down the dropdown box, ckick on the entry that you want to delete, and hit SHIFT - DELETE. If you want to delete all entries, you can just hold the keys down and it will delete them rapidly.
Sorry, I’m assuming that you are using the Google toolbar, of course.
To clear them all, you can also go to:
Tools > Options > Privacy > Saved Form Information
and click the “clear” button.