Big telephotos like the 300/2.8 already have "built-in" teleconverter in the rear group, this is why they are called "tele-photo" lenses rather than long focal length lenses. The converter keeps the lens more compact, consider the size of the "telephoto" AFS 800/5.6 (only 461mm long) compared to the "long focus" 800/8 ED (684mm long with focus unit). Actually the 800/8 ED physical length is considerably shorter than the focal length, so it is also a telephoto, but not to the same degree as the AFS 800/5.6.
So a 300/2.8 is probably more like a 200mm or 250mm lens if you ignore the rear converter group. Adding another TC to a telephoto seems like an inefficient way of increasing the focal length further since you end up with two converter groups at the rear which are not necessarily well matched. Surely it would be better to remove the rear converter group and replace it entirely with another (stronger) dedicated converter group. Better but not so practical. This is in effect what Leica did with their modular SLR telephoto system, where a single lens head can be combined with three focus/teleconverter units (see https://www.apotelyt.com/photo-lens/leica-apo-telyt-r-module).
I sometimes wonder why there are no telephotos with a variable strength rear converter group, in effect turning it into a variable aperture zoom. For example a 300-420/2.8-4 or 300-600/2.8-5.6. The AI-P 1200-1700/5.6-8 is the only example that matches this description. It is probably something like a 850/4 prime with a built-in variable 1.4-2x converter. Yes the zoom range is small and zooming in causes the aperture speed to drop off more quickly than most zooms, but it is no different to adding a separate TC, and it is certainly far more convenient, and could be properly optimised for each lens.
Thanks for this interesting information. The 300 f2.8G is my core telephoto :-)
The in-built TC in a 180-400 f4 does have its appeal. But 3.5kg is pushing the envelope for handheld shooting. Personally, I find 2.9/3.0kg the upper limit to not only shoot but carry - Thus the 200 f2G or 300 f2.8G with a D850 or D500 is the best compromise. Being shorter length, both these lenses are easier to wield than longer telephotos. Time and again they deliver in situations on birds and mammals where monopod or a tripod is impossible. Obviously, one can exploit the legendary bokeh of both these primes... full open if necessary :-)
After months of usage, I find the TC2 III and TC14 II essential to capture birds with these 2 Nikkors. The TC2 gives excellent IQ on the D850. The pay off in twinning TCs with the exotic Nikkor primes up to 600 is widely confirmed - cf Brad Hill, Thom Hogan, reviews on Photography Life etc
I still see the 500 fE FL is the optimal solution for my needs demanding longer reach on wildlife. At <3.1 kg it's manageable to carry and handhold. Less weight and longer FL than all the 400 solutions. The 500 f4 with the 1.4 TC = 700 f5.6 + the capability for much rarer cases demanding a 1000 f8, which then demands a tripod. As prices compare (RRP), 1000 less quid for a new 500 f4E is substantial. For only 1 lens, I would invest that in a 600 f4E at 300g heavier than the 180-400.
The 400 solution that would be a boon for simpler carry is a 400 fE PF to complement the 300 f4E PF. Since the excellent 400 f5.6AIS, weighing 1.2kg, Nikon have neglected this vacant niche for too long. Fresnel technology, fluorite elements, and the lighter Mg chassis of modern primes allows for a handy 400 f4E PF. Its design should maintain decent IQ with the TC1.4.... just my recurring plea to Nikon.....
Zooms have undeniable advantages, which is why so many of us depend on them. But IMHO (and African experiences since 1983) a FL of 400mm is still too short for too many wildlife subjects - especially on FX. Even with the remarkable benefits promised in a dedicated, in built TC, 560mm is also limiting for smaller subjects, pertinently birds. Either a 400 prime and 180/200-400 Zoom lacks the capacity to extend the FL to the 700 f5.6 of a 500 f4. As with the 200-400 f4, this 180-400 f4 sounds ideal in a vehicle or hide, where it's simpler to have it on a 2nd camera & even better on a D500. It will very likely become a most reliable optic in many sports arenas.
2 examples appended 300 f2.8G with TC2 III. Blue Cranes - D500 ISO400, Cape Batis with moth 100%crop - D850 ISO5600