of the cell. During prometaphase, the nuclear envelope that encloses the nucleus breaks down, and the nucleus is no longer separated from the cytoplasm. Protein formations called kinetochores form ...
Red blood cells for example, have no nucleus, so they can hold more oxygen. They're also biconcave or disc-shaped to absorb oxygen more quickly and rounded to flow easily through tiny capillaries.
After all, there must have been a time when the ancestor of eukaryotes did not possess a nucleus; the question is whether that cell possessed a mitochondrion or not.
With the exception of red blood cells, which contain no nucleus and no nuclear DNA, every one of these cells contains the human genome -- a string of three billion A's, C's, G's, and T's.