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

As with many questions about extremes, the answer depends on the precise rules you impose.

Hops count as multiple flights: 13 flights. It’s reasonable to argue that one should be very permissive when finding record itineraries. As such, the following 13-flight itinerary from SVR to SRV (a dyslexic’s nightmare?) is one of the best single answers I know:

SVR THU NAQ JUV JAV SFJ KEF ANC ANI CHU CKD RDV SLQ SRV

1. Fly from SVR Savissivik Heliport to THU Thule Air Base/Pituffik Airport on Air Greenland 9002 (Bell 212)
2. Fly to NAQ Qaanaaq on Air Greenland 9003 (Bell 212)
3. Fly to JUV Upernavik on Air Greenland 629 (DHC-8)
4. Fly to JAV Ilulissat on Air Greenland 205 (DHC-8)
5. Fly to SFJ Kangerlussuaq on Air Greenland 571 (DHC-8)
6. Fly to KEF Keflavík on Air Iceland 5492 (DHC-8) [seasonal]
7. Fly to ANC Anchorage on Icelandair 679 (Boeing 757) [seasonal]
8. Fly to ANI Aniak on Ravn Alaska 816 (DHC-8)
9. Fly to CHU Chuathbaluk on Ravn Alaska 3402 (Cessna)
10. Fly to CKD Crooked Creek on Ravn Alaska 3402 (Cessna)
11. Fly to RDV Red Devil on Ravn Alaska 3402 (Cessna)
12. Fly to SLQ Sleetmute on Ravn Alaska 3402 (Cessna)
13. Finally fly to SRV Stony River on Ravn Alaska 3402 (Cessna)

Note that many of these flights are only offered one or two days a week, so the entire route is not particularly achievable together. Google Flights is aware of all but the first flight, but doesn’t let you book many of them; nonetheless, all of the flights can be booked online individually. Note that this itinerary was shortened in the comments from 14 legs by using the seasonal flights in the middle; outside of the summer, it likely takes three legs to get from SFJ to ANC (eg, via CPH and LAX).

A quirk of this example is passing through Thule Air Base, which requires an access permit. An alternative is to fly from QUV Aappilattoq Heliport near the southern tip of Greenland, though now Google Flights is unaware of the initial four flights (all on helicopters):

QUV QFN JNN JJU UAK GOH KEF ANC ANI CHU CKD RDV SLQ SRV

Another quirk is its sensitivity to the direction of travel. An alternative is KBW Chignik Bay, a five-hop trip on Grant Aviation either way:

KBW KCL KCQ KPV PTH AKN ANC KEF SFJ UAK JJU JNN QFN QUV
KBW KCL KCQ KPV PTH AKN ANC KEF SFJ JAV JUV NAQ THU SVR
QUV QFN JNN JJU UAK GOH KEF ANC AKN PTH KPV KCQ KCL KBW
SVR THU NAQ JUV JAV SFJ KEF ANC AKN PTH KPV KCQ KCL KBW

If you forbid helicopters, then it seems one can replace the Greenlandic side with airplane flights on Air Inuit to/from YZG Salluit (one of the northernmost Inuit communities) as suggested in another answer:

KBW KCL KCQ KPV PTH AKN ANC DEN YUL YGL YPX AKV YIK YZG
YZG YIK AKV YPX YGL YUL DEN ANC AKN PTH KPV KCQ KCL KBW
YZG YIK AKV YPX YGL YUL DEN ANC ANI CHU CKD RDV SLQ SRV

In other words, KBW ↔ YZG is the only pair I know of which requires 13 flights in either direction, all on fixed-wing aircraft and all individually known by Google flights. But KBW requires a seaplane and counting multiple hops as single flights, the trip can be done (with even more hopping) in a mere five flights! (YZG → SRV only works in one direction, but requires neither a helicopter nor a seaplane.)

Hops count as multiple flights, available together on Google Flights: 11 flights. I think there are many of these. Here’s one from GTO Jalaluddin (Indonesia) to SRV, nearly 6000 miles geodesic distance:

GTO UPG DPS NRT SEA ANC ANI CHU CKD RDV SLQ SRV

Hops count as one flight: 11 flights. If you allow helicopters, I think there are many such itineraries, although they all share one half:

AUY TAH VLI BNE BKK CPH SFJ JAV JUV NAQ THU SVR
SVR THU NAQ JUV JAV SFJ KEF YUL MIA LPB TDD RIB

If you forbid Thule Air Base specifically, then you drop down to 9 flights by switching to southern Greenland (you lose two flights because of “helicopter-hopping”):

AUY TAH VLI BNE BKK CPH SFJ UAK JNN QUV

If you forbid helicopters entirely, you can achieve the same result of 9 flights by chopping off the end of the SVR itineraries:

AUY TAH VLI BNE BKK CPH SFJ JAV JUV NAQ

Hops count as one flight, available on Google Flights: 8 flights. If you only demand that Google Flights be aware of the individual flights, then you can cut off SVR from the trips in the previous category to get 10 flights (9 if you forbid helicopters).

If you want to be able to book the flights, then I can’t get any more than 8 flights … and I can’t even get them all at once. For example, for the following itineraries I can get a ticket for the first 7 flights, but I have to get the last segment separately:

AUY TAH VLI AKL IAH YYZ YYT YYR YDP
FTA TAH VLI NAN LAX YYZ YYT YYR YSO

OpenFlights (August 6, 2014 data refresh): 14 flights. I also computed the answer according to the OpenFlights data suggested in the comments. There are 9 city pairs that require 14 flights, but none seem to hold up to inspection. I include them here for completeness:

KCG KCL KCQ KPV AKN ANC LAX CPH SFJ UAK JJU JNN XEQ QUV QFN
LPS WSX DHB RCE FBS LKE SEA KEF GOH UAK JJU JNN XEQ QUV QFN
XEQ QUV QFN JNN JJU UAK GOH KEF BRU FIH FKI GOM BNC BUX IRP
XEQ QUV QFN JNN JJU UAK GOH KEF DEN ANC AKN KCG KCL KCQ KPV
XEQ QUV QFN JNN JJU UAK GOH KEF DEN ANC ANI CHU CKD SLQ SRV
XEQ QUV QFN JNN JJU UAK GOH KEF YYZ YTS YMO YFA ZKE YAT YPO
XEQ QUV QFN JNN JJU UAK GOH KEF AMS YUL YGL YPX AKV YIK YZG
YPO YAT ZKE YFA YMO YTS YYZ CPH SFJ UAK JJU JNN XEQ QUV QFN
YZG YIK AKV YPX YGL YUL JFK CPH SFJ UAK JJU JNN XEQ QUV QFN

OpenFlights (November 3, 2016 live data): 12 flights. Using a computer (as I have) helps explore the search space quickly but is sensitive to peculiarities of the data. For example, if I use the live data from OpenFlights instead, the maximum distance is now two flights shorter. Here are some of the 19 such city pairs:

BVI BEU BQL ISA BNE BKK FRA YHZ YYR YRG YMN YSO YHO
STZ SXO GRP MQH BSB ATL BOS YHZ YYR YRG YMN YSO YHO
THU NAQ JUV JAV GOH KEF AMS YUL YZV YNA YHR YIF ZLT

Believe it or not, the answer is pretty easy to find. There is a travel industry publication called the OAG. It comes in domestic and international editions. Between the two, at the back of each, there is a listing of each flight operated worldwide.
Just scan the listings until you find the longest line(s) for a particular flight. That’s it!

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.

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?

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.

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…