nothing is ever essential, however, lots of things are useful, like 4wd. Extra clearance and entry/exit angles more useful. The exhaust retarder was really useful (don't leave home without one).
Our "expedition" was 34 countries from Malaysia to Europe and back. The 5km of "difficult dunes" in SE Morocco (I know they were difficult because it said so on the map) were easy with 4wd and would have been impassable in 2wd.
The tracks in Mongolia were generally 2wd, except the very occasional 4wd bit. If it had rained 4wd would have been essential.
The east of Albania needed 4wd a couple of times.
Water across a track leaving a flooded Murray River in Aus needed 4wd and clearance - ranger in Landcruiser coming other way reckoned he'd had a bit of difficulty but "you'll be alright in that".
Hard Knott Pass, gradient 1 in 3, in English Lakes District on wet weal. There's an awkward corner on the west side. 4wd gave us reliable traction to shunt a couple of times downhill - trying to reverse uphill in 2wd would have been hairy.
Sweden driving on the half formed track through 200m of soggy landslide debris needed 4wd, but if we'd waited a day the road would have been clear.
Awkward hill in Morrocco at north end of Plage Blanche that's known for stopping people needed 4wd.
Times through Russia and Ukraine when we camped in woods that would have been inaccessible without 4wd. And the couple of lunch stops where we learned how greasy black soil can be.
Plus the many campsites where we were able to push the boundaries a bit, with 4wd in reserve to help get us out if we pushed too far.
We also carried snow chains for mud and anything else. Its been suggested that chains and 2wd provides more extra traction than 4wd. We had 4 chains which we didn't use. We had maxtrax which we used twice.
In retrospect we think our travels would have been a tad constrained without 4wd. More constrained without clearance and good entry/exit angles.
Basically though, as others have said, it depends what you want to do with the vehicle.
Having said that it was surprising where we got to in a 2-speed automatic Holden sedan (with 6 cyl 186 motor and 400,000 miles on clock) around Aus in 1978. Unstoppable.