The "best" of what the world has to offer is a very subjective criterion. For example, I have been to London several times because I have friends there, and frankly, I can live without it. If they weren't there, I wouldn't go. Same with Sydney. I have been and I have absolutely no desire to return. Obviously Unterwegs would disagree (along with a lot of other people). But I would agree that Bangkok, NY, and San Francisco are three places that are worth going to. Under your time limitations, these are my personal suggestions. In Europe, go somewhere different. Morocco, the middle east, Egypt, Spain, Russia, whatever. North America, New York, San Francisco, or Los Angeles. Asia, Bangkok, Bali, or Tokyo. I don't think I would add another continent because it's going to be a day there and a day back, which is just going to eat into your time.

Maybe something like this:

CPH-LHR-CAI-LHR-NYC-SFO-HKG-NRT-HKG-DPS-HKG-BKK-LHR-CPH

There is one Asian segment that you would have to pay a little extra for. Three days in Cairo, two in New York, three in San Francisco, three in Tokyo, four in Bali, three in Bangkok. In my opinion, that gives you three *very* different faces of Asia.