Take this line var jmax = $("ul#cyclelist li").length -1; outside the function. Else every time cycleThru is called it will create new j

    $(document).ready(function() {
      var jmax = $("ul#cyclelist li").length - 1;

      var j = 0;
      var delay = 9000; //millisecond delay between cycles
      function cycleThru() {

        $("ul#cyclelist li:eq(" + j + ")")
          .animate({
            "opacity": "1"
          }, 400)
          .animate({
            "opacity": "1"
          }, delay)
          .animate({
            "opacity": "0"
          }, 400, function() {
            if (j !== max) {
              j++;
              cycleThru();
            }
          });
      };

      cycleThru();

    });

An easy way to solve this would be to accept a cycles parameter in cycleThru, which is used to specify the amount of cycles after which recursion is stopped.

function cycleThru(cycles = Infinity) {

You then add a guard clause to the recursive function to stop the recursion.

if (cycles <= 0) return;

When you recursively call cycleThru reduce the cycles by one.

cycleThru(cycles - 1);

Finally pass the amount of cycles with the initial cycleThru call.

cycleThru(10);

Resulting in the following code:

$(document).ready(function () {
  var j = 0;
  var delay = 9000; // millisecond delay between cycles

  function cycleThru(cycles = Infinity) {
    if (cycles <= 0) return;

    var jmax = $("ul#cyclelist li").length - 1;
    $("ul#cyclelist li:eq(" + j + ")")
      .animate({"opacity" : "1"} ,400)
      .animate({"opacity" : "1"}, delay)
      .animate({"opacity" : "0"}, 400, function () {
        (j == jmax) ? j=0 : j++; // or: j = (j + 1) % jmax
        cycleThru(cycles - 1);
      });
  };

  cycleThru(10);
});