I'm surprised why no one suggested to use the WinForms Textbox.
XAML:
<WindowsFormsHost Margin="10" Width="70">
<wf:TextBox x:Name="textbox1"/>
</WindowsFormsHost>
Also don't forget the Winforms Namespace:
xmlns:wf="clr-namespace:System.Windows.Forms;assembly=System.Windows.Forms"
C#:
AutoCompleteStringCollection stringCollection = new AutoCompleteStringCollection(){"String 1", "String 2", "etc..."};
textbox1.AutoCompleteMode = AutoCompleteMode.SuggestAppend;
textbox1.AutoCompleteSource = AutoCompleteSource.CustomSource;
textbox1.AutoCompleteCustomSource = stringCollection;
With Autocomplete, you need to do it in the Code behind, because for some reasons, others might can explain, it throws an exception.