What commercial flight route requires the most (minimum possible) stops/layovers from A to B?

The United Island Hopper is probably a good candidate for that. MLL (in Alaska) to KSA (in Micronesia) will require 7 stops (at least according to Google):

For a single ticket I'm going to go with six segments, with the caveat that you shouldn't book it:

PQS-AER or PQS-VOG

Pilot Station, Alaska, requires two flight segments just to get to Anchorage. The first, PQS-BET, is a half hour flight on a tiny little Cessna, and if you haven't flown one of these, it's a bit of an adventure. After ANC you've got two more segments to get to Heathrow, a fifth segment to Moscow, and finally you arrive in Sochi or Volgograd on your sixth flight two days later in desperate need of a shower.

But don't book this itinerary. You'll save about USD 2000 (!!!) by buying PQS-BET separately, and then flying BET-AER. You'll also get access to a shorter routing that isn't available when the PQS-BET segment is included, bringing you down to five flight segments.

There are numerous places in Africa where you might require six segments to go from the same starting point, with the caveats that they're entirely unbookable with the PQS-BET segment, and that your final segment might be unbookable through global travel agents; you might have to book directly with the airline or a local charter company. Such is the case if you travel to various places in Zambia, for instance; once you get to Livingston you're on your own.

From a few directed searches I wasn't able to come up with anything requiring seven or more segments that you can actually book, single ticket or separately, that I didn't eventually find a shorter routing. Perhaps someone else will...

I could not find any flight search engine that indexes the Air Inuit airline, but given their destination map, getting from Salluit Airport, QC, Canada (YZG) (only served by Air Inuit), it would take at least 3 stops to get to Kuujjuaq.

Then, according to Google Flights, getting from Kuujjuaq, QC (YVP) to Taloyoak (YYH) takes a minimum of 4 stops. For example, one can stop in Montreal, Calgary, Yellowknife, Kugaaruk, to end in Taloyoak, NU, Canada.

So this makes a total of 8 stops from Salluit, QC to Taloyoak, NU, without even leaving Canada. There are probably even more remote communities in Canada that can be reached by even longer flights with many stops, the schedules are probably hard to find.

If you just count takeoffs and landings, Papa Westray (PPW) in the Orkney islands would be a good destination, though Google Flights can't route you there.

Nearest I can get (via the current starting point, Pilot Station) via Google Flights is PQS to INV (Inverness) - 5 flights, 4 stops.

Then Loganair will get you to Kirkwall (6th flight) - operated as FlyBE at the moment, I think. Note that Edinburgh or Glasgow (EDI,GLA) would be alternatives to Inverness but Google Flights is reporting the same hops from PQS each way.

From Kirkwall you take Loganair's inter-island flights, which form a circular route (on the same plane), with Papa Westray being 4 further flights one way, or 5 the other add one further flight for a total of 7 flights.

Note that the Westray to Papa Westray flight has its own place in the record books, and can be viewed in its entirety if you have 2 minutes to spare.

EDIT : Either MLL (Marshall,AK) or KSA (Kosrae, Micronesia) from the other answer will push the route to INV up to 5 stops, getting to Papa Westray in a minimum of 11 8 flights.

EDIT again : I must apologise for an incorrect reading of the inter-island flight timetable (or hazy memory) - in fact it isn't a full circle route; each flight covers a subset of all the islands, you ran get to Papa Westray directly from Kirkwall, so this reduces to only 8 flights. My apologies for unintentionally misleading.

Not sure about the details but flights I've taken that are multistop: Iquitos: Some Military changover in the Amazon : Float plane to the TriFrontera with Colombia and Brazil To get to Iqitos you'd have to fly from Lima, so from somewhere obscure to get to Lima you'd up the count.

Island hopping you could look at the San Blas Islands chain on the caribbean side of Panama - we hopped a few islands/stops down before getting off and the plane was continuing; I'm not sure a brief stop on a strip of tarmac counts as a stop over?