Long unresupplied stretches: I think that a person going on a two or three week trip might find it interesting (I won't say 'fun') to "try it out" once. Or of course, might find it preferable to the logistic hassles in limited cases.
My experience with long distance hikers, however, suggests that along the way most come to think of their trip in terms of how far it is to the next resupply point, and they very much look forward to the town comforts of stopping for a food resupply: taking a shower, sleeping in a bed, eating as much as possible in restaurants or fast food places, etc. Put another way, I think that a typical person starting a long distance trip would quickly have enough of trying to maximize time between resupplies. There are exceptions, but I think it would be true of most people (even the self-selected subset that read forums like this).
Longest unresupplied stretch for me: I think 8 days. That was the longest this year, and I don't believe I exceeded that on any prior trips. One factor is that thru-hikers eat a great deal, so that a "days worth" of food is heavier than for someone just starting a trip. If your base weight is reasonably low to start with AND you have thru-hiker hunger, each day of food added is a pretty significant percentage increase. It's fine to say "but I'll eat it down as I go"; it can still hurt (and literally hurt you) for the first two or three days out. And the weight never seems to decrease at the rate that you think that it would in a just universe! :-)
Another factor in long unresupplied stretches is the particular terrain you have to go through; start out carrying a heavy load, you really don't want to be bushwhacking or scrambling, or dealing with large elevation gains (or losses), all perhaps on lousy trail quality or in snow --- but sometimes that's the way the cards are dealt.
About foraging, I'll just suggest that in the vast number of real-world cases, it's impractical for anyone looking to hike a significant number of miles per day. The topic certainly has been discussed quite a lot before (don't recall if here on BPL or elsewhere, or most likely on multiple forums).