The reason your textbox needs more space is the way its control template is defined.
This is how the default control template looks like (from MSDN):
<ControlTemplate TargetType="{x:Type TextBoxBase}">
<Border Name="Border"
BorderThickness="1"
CornerRadius="2"
Padding="2">
<Border.Background>
<SolidColorBrush Color="{DynamicResource ControlLightColor}" />
</Border.Background>
<Border.BorderBrush>
<SolidColorBrush Color="{DynamicResource BorderMediumColor}" />
</Border.BorderBrush>
<!-- VisualStateManager code -->
<ScrollViewer x:Name="PART_ContentHost" Margin="0" />
</Border>
</ControlTemplate>
Studying the default control template or defining your own helps determining exactly how much extra space do you need to allocate to your text box.