SpaceX have claimed (reported here) that the Falcon 9 Block 5 first stage should be reusable up to 10 times without significant refurbishment, and up to 100 times in total. What factors are expected to limit these two numbers to 10 and 100? In other words, what is expected to need refurbishing after 10 flights, what exactly is it that is limited to 100 flights and why?
For starters, no Falcon 9 first stage has flown more than 5 times as of today. It seems that the number of 10 flights (or 11 flights if actually 10 "reuses" are meant, as stated in the question) was a parameter at least in the pre-design phase that for some reasons had to be compromised.
I am unaware of any official information about the bottleneck for reusability. However, in my view the most likely explanation is engine wear.
The engines are subject to extreme loads during firing, with very high gradients of both temperature and pressure. Combined with the intensity of the occurring vibrations, the materials that make up the combustion chamber, its cooling channels, and the nozzle experience significant fatigue. For example, hairline cracks are typical of such high intensity multi-cycle loading conditions. I will explain below what I mean by multi-cycle loading.
To answer your question explicitly, I guess it is the engine that would need significant refurbishing - if not replacement - after 10 flights. I do not believe that a metal built engine can reach a number of 100 flights. If I had to speculate where this number came from, I'd guess this has to do with the next candidates for multi-cycle fatigue, which would be the tanks for the cryogenic propellants, I guess, as they are also subject to severe conditions repeatedly.
Most definitely, a correct but very unspecific (and standard) answer to your question is that at some point (currently 5 flights of a single booster) it is not economically beneficial to go through refurbishment.
What is the loading cycle of an engine?
After production (Hawthorne, CA), the engines are shipped for testing (McGregor, TX), then shipped back for assembly (Hawthorne), after which the whole stage is shipped first for testing (McGregor), and then - finally - to the launch site (Vandenberg, CA or Space Coast, FL). Before liftoff, there's at least one static firing of all engines.
For landing, at least some engines perform a boost-back burn, a re-entry burn, and a landing burn. Note that depending on the landing mode, some of these burns are omitted. If the stage landed at sea, it is exposed to all kinds of weather for about two days, including salty air and even gusts of salt water.
I'm not sure about the exact procedure for refurbishment, but they definitely take the whole thing at least once more to McGregor for testing.
What is the loading cycle of a propellant tank?
The tanks are (at least partially) filled and drained once at the test stand, once for static firing, and once per flight. There are possibly more repetitions of this in case of scrubs.
With each filling, the tanks are exposed to cryogenic temperatures on the inside and ambient temperature on the outside. During flights, the vibrations and the hot engine exhaust contribute their share to the wear.