If the 'JavaScript served in header' option is set to Off in the General Configuration, autocomplete does not work and anything to do with the Links Browser breaks (the Insert Link button does not load in the WYSIWYG editor and the Insert > Link, Insert > Attachment and the Insert Attachment buttons do nothing).

Furthermore, the sidebar for the General Configuration screen recommends to set the 'JavaScript served in header' option to Off to increase page load performance. It warns that it might disable some plugins, but doesn't mention which ones or that autocomplete/the Links Browser will be affected.