THE CRITICAL QUESTION

"Other than the nucleus, what is the most essential organelle in a eukaryotic cell?"
I believe that other than the nucleus, the mitochondria in a cell is the most important organelle in a eukaryotic cell. The role of the mitochondria is to produce adenosine triphosphate (ATP), the cells chemical energy. Mitochondria contains matrix, soluble enzymes that catalyze small organic molecules, and cristae; internal compartments formed in the mitochondrion from inner membrane. They are studded with proteins, including ATP synthase (important enzymes that provide energy for the cell to use through the synthesis of  ATP's.)The mitochondria also controls cell growth.

Interrelationships between organelles...

The interrelationships between organelles are all connected by the endomembrane system. The endomembrane system according to Wikipedia is "composed of the different membranes that are suspended in the cytoplasm within a eukaryotic cell. These membranes divide the cell into functional and structural compartments, or organelles" This means that all organelles depend on each other to for example produce protein that is transferred from organelles then transported in vesicles from the Golgi apparatus to its destination in the body.

Which functions the cell could do without:

All cell organelles and their functions are equally important. If the nucleus handles the genetic constitution of the organism, the mitochondria is concerned with storage of food and release of energy, and for each cells role, there is a domino effect in the cell. The cell works as a whole unit and needs each function to support an organism.

Functions of organelles in the cell

What Organelles are the most critical for cell survival?

All organelles in a cell, belong there and serve a purpose, without a one major organelle the entire cell would function differently. Each and every cell in the body is like a machine that needs to function with every piece of machinery and enough workers to get the job done. Although I do believe that the nucleus is absolutely essential to a cell; without it, the cell's life would cease to exist. It stores hereditary information in DNA, and it synthesizes RNA and Ribosomes.The nucleus is basically the control center of the cell, like the brain to our bodies.