We are using a 4-continent RTW in July-Aug to visit the Great Barrier Reef and Red Center(re) of Australia, followed by visits to nature reserves other wonders in southern Africa (Victoria Falls and national parks/game reserves in Botswana and So. Africa.)
July/Aug happens to be favorable times to visit both, since N. Queensland is more comfortable then, and no stingers apparently, and in southern Africa winter is the dry season (in these areas) so in addition to the temperature being more comfortable, the game viewing is best because the animals come to the water and are not obscured by the tall brush. Game lodge/"safari" accommodation runs from basic to highly luxurious and all-inclusive (from <US$100 to well over US$1000 per person per day); in fact a number of Conde Nast's top hotels in the world last year were safari-type accommodations in Africa.
A 4-continent DONE or AONE comprising NA-Asia-Africa-Europe-NA or NA-SWP-Africa-Europe-NA would seem to fill the bill, but of course preferences enter into it. For example, DFW - YVR (enjoy a couple of days in beautiful Vancouver) - HKG - JNB - VFA (Victoria Falls) - JNB (then a few days around Kruger NP or the Sabi Sands reserves on the border) - CPT (midwinter so not the best weather probably but still okay) - London - Home allows for a lot of quality time in CX and BA F or J; or DFW - LAX - SYD - CNS (Cairns for the GBR) - JNB - LHR - Home... either would touch a couple of the world's greatest cities, some of the remaining natural wonders of the world, and could be accomplished in style without taking more than a month. Most of the longhaul flights are overnight anyway, and going west helps (us at least) with time changes, so an efficient way of spending one's time too.
At the risk being named a heretic, though, and in thinking about your aims, I'd also have a look at the SQ/NZ/VS Escapade RTW product in business class, which is considerably cheaper for ex-USA departures and throws in SQ J and Virgin Upper Class into the mix, as well as NZ's new VS-style suites in their new 2-class configuration rolling out this summer.
DFW - LAX - AKL (NZ) -SYD (NZ) - SIN (SQ) - JNB (SQ) - LHR (VS) -LAX (VS or NZ) is a pretty attractive trip, and while not much use to OW mileage addicts would probably get you to Star Silver, i.e., automatically eligible for UA Economy+ seats rather than LRTC once you return to reality. Likewise you could go LAX - HKG - SIN - CPT - JNB - LHR - LAX (no Oz/NZ) or, with no Africa, LAX - PPT or NAN (Tahiti or Fiji) - AKL - HKG - SIN - CDG - LHR (BD - small supplement) - LAX... the list is very long indeed, and very price competitive.
There certainly would be no shortage of high-end accommodations in any of these places to make this into a pretty memorable trip.
Happy planning...