Any OM/A i have read did have something to that effect. However, not as a hard rule, more as a guideline.
My current one says this:
If, in marginal weather conditions, two go arounds have been carried out at an aerodrome, consideration should be given to diverting or holding until an improvement in weather occurs.
The decision to attempt a third approach immediately should normally only be made if a significant improvement in the weather conditions has been reported or observed from the flight deck.
The options of diverting or holding will be affected by fuel considerations and by forecast weather conditions at the destination and at the alternate.
Go arounds, especially multiple go arounds, can be distressing to passengers. Commanders should, when workload permits, use the PA to give reassurance appropriate to the situation.
Personally i was only once in that situation, and we decided to divert after the second go around, although the competitor airline landed quite happily (they used 737 classics without predictive windshear system). A nearby airport had a better terrain profile and a runway more into the wind and the approach and landing there was uneventful. The real struggle came afterwards, as the company wanted us to fly back to our planned destination regardless, and it took quite a lot of talking to change their minds. Without any change in weather conditions more go arounds would have been guaranteed, as the PWS call out "windshear ahead, go around" means a go around is mandatory.