So the problem you have, is that each run of the player is ‘stand alone’ it isn’t logging what the previous run did.
You could have a user manually input a start number?
Or you could automate that, by having each run read a .txt. file, containing a number starting at 0 and overwrite the file each time with n+1? (up to a defined maximum amount).
Alternatively, perhaps you could run all the sessions in a single graph, but have a pop-up for users which breaks it up through the run? It is hard to know without some more context…
If you want a simpler way to do it one by one, and if your “items” are Revit elements, you could use a parameter. Take your list of elements, filter by Parameter equals 1, take the top item. At the end of the script, Set your Parameter of that top item to 1. When you run the script again, it will take the next top item because the previous will be filtered out.
You can also make a “reset” script that will set all the parameters to 0 again.