Go to Format / Option Inspector
Show Option Values: change from Selection to Global Preferences
Go to Global Options / System Configurations / ScreenResolutionCompatibilityMode - False
The Default.nb styles define them up to "Subsubsubsubsection" so you have two more levels to go.
They are not included in the Format>Style menu but you can use shortcuts. Hit Tab/Backspace to navigate down/up in section level. (coursor needs to be a the beginning of the cell).
If you need more you can define/edit a custom stylesheet and add ...
I figured out the answer myself. The problem can be solved by editing the stylesheet with a text editor.
To be specific:
Backup the original WolframAlphaNotebook.nb.
Open the WolframAlphaNotebook.nb with any text editor you like, search for TraditionalForm, and you'll see the following segment: