Cells die and are replaced, so over time owuld all the cells be replaced? Wouldn't the only original cells in an old person be brain cells, and all the others replaced?

yes?

Nerve cells cannot "reproduce" themselves

So when your old you only have the same nerve cells? The heart and other organs have all been slowly replaced?

Pretty much. Things are replaced at different rates though. Your entire skeleton will be replaced about four times throughout your life, whereas the lining of your mouth is completely replaced about four times a day. The molecules making up nerve cells are replaced over time, so in a sense, nerve cells do regenerate, although technically, the cell remains the same cell, however, the material your teeth are made of is never replaced, so once they're gone, they're gone.

Is there anything else that is never replaced??

I'm not sure to be honest. I have a feeling that the material making up the lens of the eye is not replaced, although I may well be wrong there.

Nerve cells cannot "reproduce" themselves

 

Although recent exiting research show that they actually can!(although not in great quantities).