In my few years of flying I had numerous engine fire warnings. The only genuine one I had was in a Bell 212 offshore Port Said on Bastille day in 1982. There was no wind, temperature above 30C and a full load of passengers, so not much power in hand. I had just rotated nose down departing from a drill ship 60 nm offshore when the No 1 engine fire warning illuminated. It became apparent that this was for real when the engine started running down towards idle. The copilot pulled the T-handle and fired the first bottle. The warning light extinguished, but after a short time re-illuminated. The second bottle was fired and the light again extinguished, though we were still less than 20 feet above the sea with airspeed around 20 knots and Nr below 85%. Then, in something like Hoffnung's Bricklayer's story, the No 2 engine chip light illuminated. My copilot , having sent a text-book Mayday call and carried out a perfect set of ECL fire drills, at this stage decided that I was going to ditch (I wasn't), leaned over and pulled the manual float inflation handle. Sadly, our Haskel booster back at base was unserviceable and float bottle pressure was a little low (though I was assured by engineering that there was enough pressure to fully inflate the floats
and the floats were only partially inflated) ......... however, I managed to nurse my wonderful 212 up to 1,000 feet and managing to achieve a magnificent 60 knots safely landed at Port Said some 1 hour later with the No 2 chip light still illuminated, having been accompanied for the last 20 miles by the Heli Union Alouette 3 (which had been on the ground undergoing maintenance at the time of our Mayday call).
The lock-nut holding the engine oil tank cap locking plate on had sheared (we found it in the filter) and the locking plate must have vibrated around to the unlocked position and fallen off, allowing engine oil to escape from the tank over the outside of the turbine. There was quite a lot of heat damage from the fire and paint had burned and blistered on the engine and gearbox cowlings. I'm rather glad I didn't have a rear-view mirror as the view may have been somewhat alarming
As one of my 'Newfie' passengers had an illicit bottle of Canadian Club whisky, he poured liberal measures of it into my post-flight coffee and the mix of champagne, ouzo and retsina at the Forasol base manager's Bastille Day party that evening meant that I woke up the following morning with an inexplicably bad headache
When I was a trainer and a sim instructor, I always tried to introduce one or two emergencies which were not covered in the ECL and required the pilots to use their technical knowledge to deal with it (or not, as was sadly too often the case).
I like oleary's rules for dealing with things not in the checklist!