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I know by changing the conflict detection settings in placement properties it's usually possible to force overlapping labels to different corners or sides of a point, but would it be possible to have all of these labels arranged in a stacked list on one side of the point without converting them to an annotation?

It would look something like this:

If it was simply 3 labels each time I could set one to top right, one to center right, and one to bottom right but there could be different number of labels for each location where an overlap occurs.

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Your best bet for automatic labels will be the maplex label engine.

Here's an article about maplex and how to enable it: http://desktop.arcgis.com/en/arcmap/10.3/map/working-with-text/enabling-maplex-for-arcgis-and-adding-the-labeling-toolbar.htm

You'll then want to set the placement properties -> position to East: enter image description here

You should also experiment with the fitting strategy tab, specifically 'key numbering'.

It may end up being impossible to get the labels to line up automatically like you have shown, but this should get you close.

Another option would be to create a process to create a new feature class with the overlapping points shifted east and spread vertically, then label the points with the labels set to display on top of the points. Set the point symbol to no color/no symbol.

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