Closing drop down tab by clicking outside.

A couple of issues

When you click the Open tab that is actually a click outside the box - so it triggers the Window.eventListener

When a click is recorded on the Tab it always does a show - so the box will never hide.

You have to do things
1. Make your click on the Open Tab toggle states on the dropdown
2. in the mouseup event handler you have to exclude clicks on the Open Tab - because otherwise the event handler will hide the box and the #notify click() eventhandler will toggle it on again

//**********Function for notification drop-down menu*******//
// SAVE REFERENCE TO OPEN TAB
var openLink = document.getElementById('note_still');
$(document).ready(function() {
    $("#notify").click(function() {
		$("#notecontainer").toggle();
    });
});
// Close the dropdown menu if the user clicks outside of it
window.addEventListener('mouseup', function(event) {
    var box = document.getElementById('notecontainer');

    // EXCLUDE CLICKS ON OPEN TAB
    if (event.target != box && event.target.parentNode != box && event.target != openLink) {
        box.style.display = 'none';
		event.stopPropagation();
    }
});
//******

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When I applied your code, I could open the tab and close it ONLY by clicking outside the dropdown, but cannot close it by clicking on the tab icon (#notify)

Here is my implementation
HTML

<div id="notify" style="display:inline-block; cursor:pointer;"><img id="note_still" src="images/note_still.png" width="31" height="30" alt="Notes">
	<div id="notecontainer" class="notecontainer" style="display:none; border:solid 1px lightslategray; margin-top:2px;">
	<div id="notedisplay" class="notedisplay">$notify_result</div>
	<div id="spanBase" style="height:40px; width:300px; background-color:gray; text-align:center;"><a style="color:white; font-weight:800;" href="">View all notifications</a></div>
</div>
</div>

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JavaScript

<script>
//**********Function for notification drop-down menu*******//
var openLink = document.getElementById('note_still');
$(document).ready(function() {
    $("#notify").click(function() {
		$("#notecontainer").toggle();
    });
});
// Close the dropdown menu if the user clicks outside of it
window.addEventListener('mouseup', function(event) {
    var box = document.getElementById('notecontainer');
	
    if (event.target != box && event.target.parentNode != box && event.target != openLink) {
        box.style.display = 'none';
		event.stopPropagation();
    }
});
//***************Ends Here*********************************//
</script>

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Working sample here

Thanks Julian for the heads-up. I made just one modification in your JavaScript solution above and then it works. I moved code line 3 to line 12 as shown below:

<script>
//**********Function for notification drop-down menu*******//

$(document).ready(function() {
    $("#notify").click(function() {
		$("#notecontainer").toggle();
    });
});
// Close the dropdown menu if the user clicks outside of it
window.addEventListener('mouseup', function(event) {
    var box = document.getElementById('notecontainer');
    var openLink = document.getElementById('note_still');
	
    if (event.target != box && event.target.parentNode != box && event.target != openLink) {
        box.style.display = 'none';
		event.stopPropagation();
    }
});
//***************Ends Here*********************************//
</script>

Select all Open in new window

That modification actually worked for me.

That modification actually worked for me.

There is no reason it wouldn't work - I am just saying it was unnecessary.

As long as the var is in a scope where the event listener can access it - it will work - but keep it on line 12 if you wish - personally I am an efficiency nut I don't like code that does unnecessary actions - and getting the element on every single mouseup is unnecessary.