What is the ruling on someone who missed a few days of last Ramadan and did not remember them until three days before Ramadan, and does not know their exact number, whether it is three or four, and he fasted three days with the intention of fasting four as a precaution, but Ramadan began before he could complete the fourth day?

You should fast for four days to ensure you have fulfilled your obligation, because one's obligation is not discharged except with certainty. So, make up the fourth day after the end of Ramadan (definition). You are not required to offer an expiation for delaying the making up of this day, as long as the delay was due to forgetfulness.