Nimgoble's is the version I used in 2015. Thought I'd put it here as this question was top of the list in google for "wpf autocomplete textbox"
Install nuget package for project in Visual Studio
Add a reference to the library in the xaml:
xmlns:behaviors="clr-namespace:WPFTextBoxAutoComplete;assembly=WPFTextBoxAutoComplete"
Create a textbox and bind the AutoCompleteBehaviour to List<String>
(TestItems):
<TextBox Text="{Binding TestText, UpdateSourceTrigger=PropertyChanged}"
behaviors:AutoCompleteBehavior.AutoCompleteItemsSource="{Binding TestItems}" />
IMHO this is much easier to get started and manage than the other options listed above.