Use a second ArrayList for the 3 strings, not a primitive array. Ie.
private List<List<String>> addresses = new ArrayList<List<String>>();
Then you can have:
ArrayList<String> singleAddress = new ArrayList<String>();
singleAddress.add("17 Fake Street");
singleAddress.add("Phoney town");
singleAddress.add("Makebelieveland");
addresses.add(singleAddress);
(I think some strange things can happen with type erasure here, but I don't think it should matter here)
If you're dead set on using a primitive array, only a minor change is required to get your example to work. As explained in other answers, the size of the array can not be included in the declaration. So changing:
private ArrayList<String[]> addresses = new ArrayList<String[3]>();
to
private ArrayList<String[]> addresses = new ArrayList<String[]>();
will work.