The caret of RadRichTextBox appears only when the text is selected. Once you try to reorder the letters within its content, it appears.

The issue is caused when ScrollViewer control has different than the default style for the corresponding theme and there are set Padding and/or Margin properties. 
Workaround: the template of the Caret should be edited and Style="{x:Null}" should be set on the ScrollViewer which is defined inside the Caret control template.