ArrayList of String Arrays

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.

I wouldn't use arrays. They're problematic for several reasons and you can't declare it in terms of a specific array size anyway. Try:

List<List<String>> addresses = new ArrayList<List<String>>();

But honestly for addresses, I'd create a class to model them.

If you were to use arrays it would be:

List<String[]> addresses = new ArrayList<String[]>();

ie you can't declare the size of the array.

Lastly, don't declare your types as concrete types in instances like this (ie for addresses). Use the interface as I've done above. This applies to member variables, return types and parameter types.

List<String[]> addresses = new ArrayList<String[]>();
String[] addressesArr  = new String[3];

addressesArr[0] = "zero";
addressesArr[1] = "one";
addressesArr[2] = "two";

addresses.add(addressesArr);

Simple and straight forward way to create ArrayList of String

    List<String> category = Arrays.asList("everton", "liverpool", "swansea", "chelsea");

Cheers