HorizontalAlignment=Stretch, MaxWidth, and Left aligned at the same time?

You can set HorizontalAlignment to Left, set your MaxWidth and then bind Width to the ActualWidth of the parent element:

<Page
  xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
  xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml">
  <StackPanel Name="Container">   
    <TextBox Background="Azure" 
    Width="{Binding ElementName=Container,Path=ActualWidth}"
    Text="Hello" HorizontalAlignment="Left" MaxWidth="200" />
  </StackPanel>
</Page>

<Grid>
    <Grid.ColumnDefinitions>
        <ColumnDefinition Width="*" MaxWidth="200"/>
    </Grid.ColumnDefinitions>

    <TextBox Background="Azure" Text="Hello" />
</Grid>