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From the rules:
Maximum two stopovers permitted in the continent of origin of which a maximum of one stopover is permitted in each direction in the country of origin.
In my example, CAI, my starting point, is in the same continent as LHR. Therefore aren't three stops in London disallowed?
That's correct; two stopovers (24h+) in the continent of origin. However there are no rules regarding where one can lay over or connect within that continent, nor any rule that you couldn't stop over twice in the same city.