Suggestions come from your address books. If they were erased you would not get suggestions. By default you have 2 address books. The Personal and the Collected.

I have erased all address books--personal and collected. I still get prompted e-mail addresses in the "To". I want these addresses gone! I thought that erasing all address books would accomplish this but it hasn't

You have established the fact you want them gone so you do not have to repeat that anymore.

Do this... Open a Write window. Enter some text in an address field to make the annoying address appear and accept it. Enter Test in the subject line Enter Test in the message body Save to Draft folder Now go to the Draft folder and Make sure the Message Pane is turned on. Pressing F8 toggles the message pane on and off. Select the Test message with a single left click. In the header of the message pane in the To field you will see the annoying address with a color filled star to the right of it. The color filled star means it is in one of your address books. Click the star to open the edit dialog box for this contact. You will get some info from that dialog box, like which book it is in or that you cannot delete this contact because it is in a mailing list. There is also a Delete button. Delete the contact. Delete the Draft message.

Repeat as needed.