note: 'person::person()' is implicitly deleted because the default definition would be ill-formed

Well, the problem is not with that "note". The "note" simply explains the reason for the error. The error is that you are trying to default-construct your person object when class person does not have a default constructor.

Instead of trying to default-construct it, you can {}- initialize that const member and the code will compile

person bob = { nextPersonID++, "Bob", {}, 1 };
bob.birthdate.day = 1;
bob.birthdate.month = 1;
bob.birthdate.year = 1990;
...

Alternatively, you can simply write your own default constructor for the class.

The problem has not to do with a "default-construct ... when class person does not have a default constructor." The problem has to do with having a constant in the declaration of the class and a constructor that does not guarantee that the constant will be defined. Suggest using an "initializer list".

struct Person {
        int id;
        string name;
        date birthdate;
        const int numberOfAddresses;
        address addresses [1];

        Person(int); // constructor declaration
        Person() : numberOfAddresses(1) {} // constructor definition.
                      // ": numberOfAddresses(1)" is the initializer list
                      // ": numberOfAddresses(1) {}" is the function body
    };
    Person::Person(int x) : numberOfAddresses(x) {} // constructor definition. ": numberOfAddresses{x}" is the initializer list
    int main()
    {
        Person Bob; // calls Person::Person()
        Person Shurunkle(10); // calls Person::Person(int)
    }