I disagree with all of you. Many cells live and function perfectly fine without a nucleus. You personally have 20 to 30 trillion of them, since human red blood cells, or erythrocytes, do not have a nucleus at maturity. Also, we must consider the prokaryotes, which by definition lack membrane-bound organelles, including the nucleus. The prokaryotes, which include all members of the domains Archaea and Bacteria, can do all the life processes, including copying DNA and manufacturing proteins. Additionally, these organisms lack mitochondria, but they have evolved many alternate ways of producing energy to power the cell, including fermentation and anaerobic respiration. Since prokaryotes have been on the planet much longer than eukaryotes, and since there are many, many different kinds of prokaryotes, we can hardly dismiss them as inconsequential in this discussion. They make it obvious that a cell can be an evolutionary success without either nucleus or mitochondria.

The one thing that all cells do have to have is a cell membrane. The cell membrane defines the cell by creating a separation between the cell itself and the world around it. The cell membrane controls what passes into and out of the cell. Cell membranes are studded with receptors, identifiers, and passageways. They are active organelles that no cell can survive without. The cell membrane is the most important component.

It is the nucleus. The nucleus controls the cell's life activities. When a nucleus is removed from the cell, the cell dies. The nucleus contains the DNA which is the blueprint for the cell. It gives information for the cell's machinery to manufacture proteins that are structural parts of the cell, enzymes and other important proteins necessary for the cell to live. DNA is the material that makes up a cell's chromosomes and genes. Therefore, without those chemical instructions, RNA couldn't copy the DNA, travel to the ribosome and manufacture necessary proteins. Although the mitochondria are important to carry out respiration in a cell, the nucleus is the most important organelle.