WJG (04/10/25) A simple enough question I suppose, with a simple solution.
set str "Line 1.\nLine 2.\nLine 3." puts [llength [split $str \n]]
If str is surrounded by whitespace, then best delete it to get a more accurate result based upon readable text.
set str "\nLine 1.\nLine 2.\nLine 3.\n" puts [llength [split [string trim $str] \n]]