This cell organelle does not contain DNA

The correct answer is Lysosomes.

To understand why lysosomes do not contain DNA, let us examine the characteristics of the cell organelles mentioned in the options:
1. Nucleus: This is the primary control center of the cell and houses the majority of the cell's genetic material in the form of DNA.
2. Mitochondria: Often called the powerhouses of the cell, mitochondria possess their own circular DNA (known as mtDNA) and ribosomes, allowing them to replicate independently of the cell.
3. Chloroplast: Found in plant cells, chloroplasts are responsible for photosynthesis and, like mitochondria, contain their own DNA (cpDNA) and ribosomes.
4. Lysosomes: These are membrane-bound organelles containing digestive enzymes used to break down waste materials and cellular debris. Unlike the nucleus, mitochondria, and chloroplasts, lysosomes do not contain any genetic material (DNA).

Therefore, among the given options, the organelle that does not contain DNA is the lysosomes.