Announcing the deal alongside Boeing's chief executive, Ray Conner, in New York, O'Leary said the bigger fleet would allow Ryanair to increase passenger numbers from 82 million now to more than 150 million by 2024.
The deal is worth up to $22bn (£13.6bn) at list prices, though airlines rarely pay the full price on big orders. Ryanair has made a firm pledge to buy 100 of the 737 Max 200 planes, with an option on another 100, and O'Leary said Ryanair had never failed to take up options on aircraft it ordered.
O'Leary said Ryanair would win business from big flag carrier airlines such as Ailtalia, Lufthansa and British Airways and extend its network to countries such as Israel and Lebanon. "You see all the failing flag carrier airlines cutting capacity and cutting routes. We intend to use these brilliant new aircraft to expand into those markets and double in size."
The order for the planes, modified to Ryanair's requirements, is Boeing's biggest from a European airline. O'Leary, who has taken a back seat at Ryanair in recent months, said the new aircraft would be a game-changer, allowing more passenger leg room as well as eight extra seats, which would increase revenues per plane by $1m a year.
O'Leary said he had been asking Boeing and its rival Airbus for years to produce a plane with about 200 seats. Boeing agreed to increase the capacity of the 737 Max 200 to 197 seats from 189 in Ryanair's existing 737-800s, taking the budget carrier to a "sweet spot" of efficiency, he said.
By keeping the number of seats below 200, Ryanair does not have to put an extra flight attendant on the plane because one is required for every 50 passengers.
As well as increasing capacity, the planes will cut fuel consumption by up to 18% and noise emissions by 40%.
"I feel like a child who has woken up on Christmas morning and Santa Claus has given him exactly what he is after," O'Leary said.
The new targets for expansion put Ryanair back on the front foot after profits fell last year and it admitted losing touch with passengers' requirements. The low-cost carrier is trying to improve its service after O'Leary acknowledged it was no longer good business to "deliberately piss people off".
It has introduced allocated seating, relaxed cabin bag restrictions, reduced charges and loosened booking conditions. Last week the airline announced that a record 9.4 million passengers flew Ryanair in August.
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