Is there such a concept as 'Max Overhauls' for aircraft components?
Yes, but its a fairly new change from the FAA. The FAA recently regulated part Life Limits, a limit in place at certification time (or through a later AD etc.). The IATA also outlines guidance for life limited parts. This is a fairly new reg and many of the older aircraft out there may not be subject to it since they were certified prior to it. This AC also outlines some info relating to it.  

However for any mechanical part there, in theory, is a limit to the amount of times it can be overhauled if that work requires reductive machining. Something like a crank case, cylinder head or piston sleeve can only be machined so many times before there is not enough material left for a proper overhaul. 

The definitions of overhaul vs rebuild are important. A rebuild may reset a parts life, you can read more on that in this question.

Any life limited part should be listed in the documentation for the specific airframe/model in question. For example here is the list of the Robinson R-22 helicopter. More notably the Cirrus SR-22 has an airframe life limit of 12,000 hours as part of its certification.


  The certification requirements of 14 CFR 23.573 require that the
  composite airframe structure, cabin, wing, empennage their carry thru,
  and attaching structure whose failure would be catastrophic must be
  designed to damage tolerant criteria. Damage tolerant certification
  for the selected airframe life of 12,000 flight hours has been
  established for all of the affected parts with no special structural
  limitations or inspections.