Is there a limit to how many hours a person can fly per month?
With respect to air carrier operations in the US, 14 CFR 121.481 requires:


  (e) No pilot may fly as a member of a crew more than 100 hours during any one calendar month.
  
  (f) No pilot may fly as a member of a crew more than 1,000 hours during any 12-calendar-month period.


14 CFR 135.265 commuter:


  (1) 1,200 hours in any calendar year.
  
  (2) 120 hours in any calendar month.


Under Part 91, fractional ops, 14 CFR 91.1059 states:


  (a) No program manager may assign any flight crewmember, and no flight crewmember may accept an assignment, for flight time as a member of a one- or two-pilot crew if that crewmember's total flight time in all commercial flying will exceed -
  
  (1) 500 hours in any calendar quarter;
  
  (2) 800 hours in any two consecutive calendar quarters;
  
  (3) 1,400 hours in any calendar year.


Otherwise, there is no direct limit under Part 91. However, if you are fatigued enough to cause a problem, FAA can still charge you under general "unsafe operation" rules.