What is the maximum amount of flight hours an aircraft can have a month?
I don't think there is any regulation on max flying hours, just regulations on maintenance intervals.

The "A" check is the most frequent check, and according to Wikipedia:


  This is performed approximately every 125 flight hours or 200–400
  cycles. It needs about 20–50 man-hours and is usually performed
  overnight at an airport gate or hangar. The actual occurrence of this
  check varies by aircraft type, the cycle count (takeoff and landing is
  considered an aircraft "cycle"), or the number of hours flown since
  the last check. The occurrence can be delayed by the airline if
  certain predetermined conditions are met.


The other checks include:


B check - every 4–6 months, requiring about 150 man-hours and is usually performed within 1–3 days at an airport hangar
C check -  every 20–24 months, requiring about 6000 man-hours, taking 1 to 2 weeks to complete
D check - major maintenance check, occuring about every 6 years, requiring 50,000 man hours, generally spanning a couple months.


Note that all of these times are approximate, depending on the exact maintenance plan filed by the airline and the aircraft type. Also cycle count may be factored into the maintenance intervals in addition to flight hours so aircraft that do many short-haul flights may have more frequent maintenance.

So in any case, only looking at the A checks, if you can land, refuel, service, and get back in the air in 2 hours between 10 hour flights, the plane could stay in the air for 20 hours/day. Every 6 days, it will need some downtime for the A checks, let's say the airline puts enough  people on it to get the check done in 5 hours. So that's 125 hours per cycle: 120 hours flying, 5 hours in maintenance. They can fit 4.83 of these cycles in a 30 day month so the aircraft could be flying up to 580 hours in any given month (but that last A check gets deferred to the next month, so that means less time in the air that month.

With longer flights (which reduces the overhead from turnaround times) and tighter turnaround times, the flight hours per month could be higher. If you really want the maximum for a month, then if you assume 125 hours per maintenance inverval and that the plane is kept in the air as long as possible with 0 turnaround time, that's around 690 flying hours out of 720 total hours in a month.

Of course, this makes a big assumption that the A check interval really is 120 hours, and that the time to do the maintenance is 5 hours, in the real world both numbers will vary based on aircraft, airline, and maintenance staffing levels.

Plus, over the longer term, you need to factor the other, less frequent but also more time consuming maintenance checks (B through D checks), as well as allow downtime for unexpected repairs.