The following snippet gives an example that shows how to get an element from a List at a specified index, and also how to use the advanced for-each loop to iterate through all elements:

    import java.util.*;

    //...

    List<String> list = new ArrayList<String>();
    list.add("Hello!");
    list.add("How are you?");

    System.out.println(list.get(0)); // prints "Hello!"

    for (String s : list) {
        System.out.println(s);
    } // prints "Hello!", "How are you?"

Note the following:

• Generic List<String> and ArrayList<String> types are used instead of raw ArrayList type.
• Variable names starts with lowercase
• list is declared as List<String>, i.e. the interface type instead of implementation type ArrayList<String>.

References

API:

• Java Collections Framework tutorial
• class ArrayList<E> implements List<E>
• interface List<E>
  • E get(int index)
    • Returns the element at the specified position in this list.

Don't use raw types

• JLS 4.8 Raw Types

  The use of raw types is allowed only as a concession to compatibility of legacy code. The use of raw types in code written after the introduction of genericity into the Java programming language is strongly discouraged. It is possible that future versions of the Java programming language will disallow the use of raw types.

• Effective Java 2nd Edition: Item 23: Don't use raw types in new code

  If you use raw types, you lose all the safety and expressiveness benefits of generics.

Prefer interfaces to implementation classes in type declarations

• Effective Java 2nd Edition: Item 52: Refer to objects by their interfaces

  [...] you should favor the use of interfaces rather than classes to refer to objects. If appropriate interface types exist, then parameters, return values, variables, and fields should all be declared using interface types.

Naming conventions

Variables: Except for variables, all instance, class, and class constants are in mixed case with a lowercase first letter.

It would be simpler if you were to just declare it as a fixed-size List (but with mutable elements) - does it have to be an ArrayList?

List<String> places = Arrays.asList("Buenos Aires", "Córdoba", "La Plata");

Or if you have only one element:

List<String> places2 = Collections.singletonList("Buenos Aires");

This would mean that places2 is immutable (trying to change it in any way will cause an UnsupportedOperationException exception to be thrown).

To make a variable-size list that is a concrete ArrayList you can create an ArrayList from the fixed-size list:

ArrayList<String> places = new ArrayList<>(Arrays.asList("Buenos Aires", "Córdoba", "La Plata"));

And import the correct package:

import java.util.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.