The takeoff limit for the B744 manual I have states the takeoff limits are available for 5 minutes with all engines operating or "
10 minutes with a loss of thrust on 1 engine during takeoff" for the CF6 or "
10 minutes provided its use is limited to situations where an engine failure occurs" for RB211.
CON limit is exactly that - continuous.
There may be some confusion with the question - as the TOGA refers to the AFDS system which in turn will control the TMC and associated thrust setting. Obviously - TOGA mode and associated thrust settings are used for takeoff and go-around.
My manuals have no specifics regarding a go-around limit with either the 2000 fpm first click of TOGA or the full Toga go-around (second click). The manual simply states that the above mentioned limits are for the "
Takeoff condition". Further research reveals that at SL ISA the Full rating takeoff thrust is 108.4%N1 for the CF6 and the go-around thrust limit is 103.8%. In addition the RR211 shows around 1.71 EPR takeoff limit and around 1.69 EPR limit for go-around from the manual.
In a practical sense - even if you apply a 5 minute all engine operating takeoff limit to the go-around thrust setting - that is a climb of around 10,000' with the first TOGA switch depress (most likely much less than full go-around thrust at landing weight in any case). Any missed approaches need a 10,000' climb?
With regards to part of the question -
Full beans is acceptable for take off and for go around, but how long should one operate at mct or less before applying toga thrust again?
- I can find no reference in the manuals for any EGT cooling period or any other restriction - on that one I have no idea..... all I can say in practical experience that it would most probably be an emergency condition that would warrant such action as a GPWS pull up or windshear recovery maneuver - in which case all bets are off. Otherwise normal operations would most probably not see a need to return to takeoff or go-around thrust having completed the takeoff or go-around maneuver.
Hope it helps to answer your question.