Are all body atoms really recycled several times during a life?
Rudolf Shoenheimer and David Rittenberg were key figures in introducing the isotopic tracer technique to biology.
This technique, which may be dated to the discovery of deuterium by Urey in 1933, produced a revolution in biological thought.
Much of the early work was done at the Department of Biological Chemistry at Columbia University under the chairmanship of H. T. Clark.
Up to this time it was thought that the components of tissues were relatively stable, and that ingested food was metabolized immediately to provide a source of energy.
The first experiments showed that when deuterated fatty acids were fed to mice, most of the label was initially deposited in adipose tissue.
It was concluded that adipose tissue acts a 'fuel depot' that is in a state of metabolic flux and is the direct source of fatty acids burned as a fuel (see here for refs). Fats present in food are not primarily used an an immediate energy source.
When a stable isotope of nitrogen became available (N15), it was found that proteins were also 'in a state of dynamic flux' and were being continuously regenerated and broken down.
The first experiments were done with N15-labeled tyrosine and it it worth quoting from the paper (Studies in Protein Metabolism. VII The metabolism of tyrosine by D Rittenberg Rudolf Schoenheimer &amp;  S. Ratner)

The results indicate that in our rat the nitrogen of at least one
amino acid, tyrosine, was only partly excreted in the urine, while
almost half of it was retained in the body proteins.
Of this deposited nitrogen only a fraction was found attached to the original
carbon chain, namely to tyrosine, while the bulk was distributed over other
nitrogenous groups of the proteins

Shortly after giving a series of lectures at Harvard University in 1941, Shoenheimer took his own life.  The lectures were published posthumously as a classic book called The Dynamic State of Body Constituents.
Further information about the early  work on isotopes, their role in changing completely the way we think about metabolism, and the role of Columbia University in shepherding in this great era, may be found in the following open-access article by Eugene. P. Kennedy:
Hitler’s Gift and the Era of Biosynthesis
Another good reference on the work of Schoenheimer may be found here

So are all proteins broken down and regenerated?  I know of any one exception and that is the lens crystallins where there is virtually no turnover
throughout the life of an individual.
The crystallins in our lens are synthesized at birth and remain with us virtually unchanged for life.
There is a very nice open-access paper on this published in PLOS One:
Radiocarbon Dating of the Human Eye Lens Crystallines Reveal Proteins without Carbon Turnover throughout Life
(Lynnerup et al. 2007)