Where do I go and how do I increase the
FontSize in this drop down menu to make it more easy to read?
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Details of the solution and additional comments may be found in: How to change the default font for all cells in all notebooks
Format / Edit Stylesheet
Core.nb(no all of them, only selected on a picture below)
(the Core.nb is in $InstallationDirectory/SystemFiles/FrontEnd/Stylesheets)
"CodeAssistUsage", the rest will inherit it.