It is not unreasonable to regard a cell as a machine that is constantly rebuilding itself - organelles and protein machinery are constantly torn down and replaced. It is also not unreasonable to regard tissue as a collection of cells that is constantly rebuilding itself: cells destroy themselves or are destroyed by watchdog systems, and new cells are created to replace them. This sort of thing happens rapidly indeed in some parts of the body, such as the blood and stomach lining, but there are portions of your nervous system where cells will never be replaced under normal circumstances - the cells you were born with are the very same cells you have now.
These long-lived cells are the most vulnerable to forms of age-related damage involving build up of metabolic waste products, and the related slow failure in the ability of cells to recycle their own damaged components. There is no fallback to replacing cells wholesale in this case, or at least not in our species, so long-lived cells must forge ahead and struggle to do their job no matter how damaged they are. The existence of these cells is a good argument for the need for in situ repair technologies, able to reverse damage and remove other hinderances in order to allow long-lived cells to regain their vigor and function - goals that are hard to attain with the present generation of cell replacement technologies emerging from the field of regenerative medicine.
Now consider this: it may be the case that some of the individual vital proteins in the machinery of long-lived cells are also never replaced. Some of your complex individual proteins, important cogs and gears in important cells, might be as old as you are. The very same sorts of concern about vulnerability surface here as well. Here is news of research in rats:
The scientists discovered that certain proteins, called extremely long-lived proteins (ELLPs), which are found on the surface of the nucleus of neurons, have a remarkably long lifespan. While the lifespan of most proteins totals two days or less, the Salk Institute researchers identified ELLPs in the rat brain that were as old as the organism. ... ELLPs make up the transport channels on the surface of the nucleus; gates that control what materials enter and exit. Their long lifespan might be an advantage if not for the wear-and-tear that these proteins experience over time. Unlike other proteins in the body, ELLPs are not replaced when they incur aberrant chemical modifications and other damage.
The fundamental defining feature of aging is an overall decline in the functional capacity of various organs such as the heart and the brain. This decline results from deterioration of the homeostasis, or internal stability, within the constituent cells of those organs. Recent research in several laboratories has linked breakdown of protein homeostasis to declining cell function. ... Most cells, but not neurons, combat functional deterioration of their protein components through the process of protein turnover, in which the potentially impaired parts of the proteins are replaced with new functional copies. Our results also suggest that nuclear pore deterioration might be a general aging mechanism leading to age-related defects in nuclear function, such as the loss of youthful gene expression programs.
Given how much longer humans live in comparison to rats, it may be that there are no proteins in the human body that never turn over. But I wouldn't be surprised to find that the situation for old humans is exactly the same as described above for old rats.