All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
The question sounds a bit contradictory. First, you say that the woman in hand has a weak memory and cannot remember the exact number of days that she has missed during previous years. Then, you mention that she starts making up for the days that she missed and makes the intention for finishing up all the days before the following Ramadan comes but that Ramadan always comes upon her before she completes them.
Your account indicates that the woman in hand was aware of the exact number of days that she missed; otherwise she would not have been certain that she had not yet finished the number of days that she still had to fast. Anyway, if there is an obligation that is due on a person, then the person should not be cleared of such an obligation until he fulfills it with certainty. Since the woman in hand is not certain about the exact number of days that she has missed, then she has to fast a number of days through which she will be confident that she certainly and completely fulfilled all that which she missed.
The second issue is about delaying the fasting that she missed in Ramadan until a following Ramadan came. If the days that were to come ahead of the beginning of Ramadan were sufficient for making up for the days that she missed in a previous Ramadan and she did not have excuses such as sickness or menstruation that prevented her from fasting, then she is required to offer an expiation for this delay. The expiation to be given is a mudd (a measure equal to 750 gm) of food for each day of fasting. She does not have to repeat the expiation with repetition of delays.
For example, let us suppose that she has to make up for ten days of fasting and there are only ten days left until Ramadan. If she did not fast those ten days during these ten days that are ahead of her and she does not have an excuse that prevents her from fasting, then she will have to offer the above mentioned expiation. This expiation should not be repeated if she delays the fasting that is due on her until a third Ramadan comes.
Allaah Knows best.