A 747 taking off at or near max gross weight will have a flaps up, minimum safe airspeed and climbout airspeed speed greater than 250 knots.
In such cases you didn't notify ATC as they expected it.
I tried to find what the exact numbers would be at 840,000 lbs, but I seem to have thrown out that manual.
I did, however, find a reference for a 747-200 with JT9D-7Q engines that gives the flaps up holding speed at 800,000 lbs as 259 kts at 5,000 feet.
Also, I do remember being on an arrival once, with the weight low enough that we could safely maintain well below 250, but that when we saw we were icing badly, we speeded up to somewhere around 300 to get the ram air temp above freezing.
In that case we notified ATC that we were doing so.
Also, ATC occasionally requests max speed.
For example, on Hajj flights into and out of Jakarta Halim airport in the late 1990s it was common practice for controllers to put you down low to get you under the traffic into Cengkareng airport.
They would then often request max speed.
aviation.stackexchange.com/users/394/Terry
FAA has letters of agreement with US DOD, which authorizes certain military aircraft to fly faster than 250 KIAS below 10,000 feet.
A good example of this would be the Northrop T-38 Talon , described in FAA JO 7610.4 Special Military Operations : 12-10-11.
AUTHORIZATIONS T-38/T-1 airspeeds in excess of 250 KIAS below 10,000 feet MSL are authorized by competent military authority in accordance with FAA Order 7110.65 and the letter of authorization granted to DOD.
The reason for this waiver is discussed in NTSB report number FTW90FA151 :
THE T-38'S SPD WAS ABOUT 330 KTS; THE SPD LIMITATION OF 250 KTS HAD BEEN WAIVED FOR T-38 ACFT, DUE TO THE ACFT'S LACK OF CONTROLLABILITY AT
THE LOWER SPD.
The relevant letter from forums.jetcareers.com : Also, from AIR FORCE INSTRUCTION 13-201 : 14 CFR Part 91.117 covers aircraft speed.
Recognizing that some DoD aircraft performance requirements exceed 250 knots, the FAA issued an exemption to 14 CFR Part 91.117.
The exemption has been claimed for other aircraft too- like F-15 Eagle, for example : ...
climb-out occurs at 350 KCAS for an air-to-air configured jet and 330 KCAS on one with air-to-ground ordnance.
...
this is significantly higher than the 14 CFR speed restriction of 250 knots below 10,000 feet.
The F-15E, like most fighter aircraft, falls under the Letter of Agreement between the DoD and FAA allowing some military aircraft a waiver to that speed limit.
aviation.stackexchange.com/users/9731/aeroalias
A number of military aircraft - one that I can cite off the top of my head is the T-38 - have speed restriction waivers from the FAA during operations for safety.
The T-38, for example, flies the traffic pattern at 270 KIAS in order to provide sufficient maneuvering capability without exceeding the AoA limitations of the aircraft and stalling it.
The F-104 probably has similar limitations in performance.
The SR-71 rotated at 230 KIAS and the pilot had to immediately retract the gear after liftoff to prevent structural damage from exceeding Vle of 300 KIAS.
As for commercial and civil aircraft, the speed restriction of 250 KIAS does not present much of an obstacle or impediment to safe operation, albeit it’s a little slow for jets.
aviation.stackexchange.com/users/15637/Carlo Felicione
777-200 stall speed at max t.o. weight is 192 kts.
1.3 buffer brings that to 252 kts.
777-300 stall speed at max t.o. weight is 207 kts.
1.3 buffer brings that to 269 kts.
Because of the higher stall speeds, especially for the 777-300, ATC knows we'll exceed 250 kts on departure and frequently acknowledges it before we mention it.
On arrival, at lighter weights, it's not an issue.
On the 777-300 you don't have to have Flaps 1 out until approx 220 kts.
at max landing weight.
On the 777-200 it's slightly lower (approx.
215 kts).
So the reduction in weight from max T.O. weight to max landing weight reduces the minimum clean speed by 35 kts (777-200) and 50 kts (777-300).
Most fighters, in normal ops with a clean wing, won't be close to stalling at 250 kts.
But they can pull more G's, thereby having more maneuverability, if they're flying faster.
aviation.stackexchange.com/users/44827/Bert Booth
In the EA-6B we had a waiver to climb out at 300KIAS.
Coming home
we were lighter and could easily make 250, but we were waivered to 300 if we were in formation to have extra margin for the wingman.
aviation.stackexchange.com/users/22182/Michael Hall
14 CFR 91.117  limits aircraft to 250 knots indicated airspeed below 10,000 feet, but 91.117(d) allows aircraft to fly a higher speed if the minimum safe speed is higher. What aircraft and aircraft conditions might require an indicated airspeed higher than 250 knots?
What aircraft fly faster than 250 knots indicated airspeed under 10,000 feet for safety reasons?
aviation.stackexchange.com
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aviation.stackexchange.com/questions/34480
aviation.stackexchange.com/users/8490/ryan1618
Many military aircraft fly at speeds greater than 250 knots. The 747 and 777 also take off at a greater speed when they are at or near max. gross weight. There are many aircraft that will exceed this limit: 747, the F-15E,  the T-38, the 777-300, the EA-6B and others have a waiver from the FAA.
There are many aircraft that will exceed this limit: 747, the F-15E,  the T-38, the 777-300, the EA-6B and others have a waiver from the FAA.