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You need intercoolers to make high power at altitude. Otherwise you are just adding weight, bulk and cost to make a slightly better low altitude engine. The supercharger adds considerable heat to the intake charge. For instance the supercharger on a Merlin XX added 148 degrees celsius (over 260 degrees F) to the temperature of the incoming air when in high gear. using more compression in the supercharger/s just adds more heat and at some point you reach the detonation point.
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Excellent post, however I'd say that intercooler was not mandatory for 2-stage engines to work well at high altitude. The gain vs. 1-stage versions of same engine was about 60% on the DB 605L vs. 605A at 35000 ft, similar for the best 1-stage V-1710 vs. the 2-stage versions. Both DB-603L (or was it LA?) and Jumo 213F were 2-stage engines without intercoolers.
Cooling drag of the Jumo 213E was about 35% greater than of the 213E, so we might take that factor in a consideration. One of things that made the otherwise unremarkable XP-40Q2 and P-63C as fast, if not faster than Spitfire IX/VIII.