The arrow operator has no inputs.  Technically, it can return whatever you want, but it should return something that either is a pointer or can become a pointer through chained -> operators.
The -> operator automatically dereferences its return value before calling its argument using the built-in pointer dereference, not operator*, so you could have the following class:
class PointerToString
{
    string a;
public:
    class PtPtS
    {
    public:
        PtPtS(PointerToString &s) : r(s) {}
        string* operator->()
        {
            std::cout << "indirect arrow\n";
            return &*r;
        }
    private:
        PointerToString & r;
    };
    PointerToString(const string &s) : a(s) {}
    PtPtS operator->()
    {
        std::cout << "arrow dereference\n";
        return *this;
    }
    string &operator*()
    {
        std::cout << "dereference\n";
        return a;
    }
};
Use it like:
PointerToString ptr(string("hello"));
string::size_type size = ptr->size();
which is converted by the compiler into:
string::size_type size = (*ptr.operator->().operator->()).size();
(with as many .operator->() as necessary to return a real pointer) and should output
arrow dereference
indirect dereference
dereference
Note, however, that you can do the following:
PointerToString::PtPtS ptr2 = ptr.operator->();
run online: https://wandbox.org/permlink/Is5kPamEMUCA9nvE
From Stroupstrup:
The transformation of the object p into the pointer p.operator->() does not depend on the member m pointed to.  That is the sense in which operator->() is a unary postfix operator.  However, there is no new syntax introduced, so a member name is still required after the ->