If you look in a Maintenance Manual, you'll find a table or graph showing the required maintenance required for an EGT exceedance, based on the time and peak temperature of the exceedance. It can be anything from no action required (or no action required for up to X occurrences) to remove engine depending on the severity of the exceedance.
Takeoff is normally limited to 5 minutes - for most engine installations 10 minute TO is only allowed in the event of an engine failure.