You can put it in SharedPreferences if it a compatible data type, or serialize the list to a file in the internal or external storage (see saving files, you probably want internal storage for your situation).
You can read the list in the onResume() method of the Activity or the onReceive() method of the receiver. Writing will occur whenever a new element needs to be added to the list, you will deserialize the list, add the element and re-serialize it. You may need to place the reading and writing in a synchronized block or method to prevent simultaneous access.
If the list may be added to outside of the Activity when the Activity is resumed you may need to provide a broadcast receiver or other mechanism for the Activity to be alterted and reload the list.
As Gabe pointed out, you could store it in a static variable somewhere, but the way you described the situation serializing sounds like a better option. Otherwise the list is gone if your process is terminated by the Android system.
As kcoppock pointed out, you could use an SQLite database to store the data, but this sounds like overkill from what you described. You will be in for some additional coding without much benefit.