It depends on when you do the query. Years ago (mid 2000s) at ITA, Carl wrote code to find the longest possible shortest route between any two airports with scheduled flights. At the time the winner was Wasu, New Guinea to Peawanuck, Ontario, which took a minimum of 90 hours and many flights. There are airports you can only get to at the end of a long line of stops; this drives the number of stops way up.

Slightly unrelated, but you can book flights between Seattle (SEA) and Everett (PAE), and there's a PAE-LAS-SFO-SEA route that takes 20 hours.

These two Airports are only 37 miles away from each other. There's a shuttle between them that takes 75 minutes.

This is now obsolete question, because of technical progress. Just now all change.

Economically, most cost effective route one hop, without additional landings. And also, it is very big hassle, to manage all passengers and their luggage in hub, and all these transfers are time consuming and tiresome.

~15 years ago, unfortunately, only largest planes (747/380) could fly any route in one hop, but smaller (ie 737) could not. And all air transport economy built with concept of "hub world" - existed several big hubs, like Frankfurt, between them flight big planes, and to reach smaller airports, used small planes or some other transport.

Also important thing, air companies in past have to maintain good relations with hubs, so this was also question of politics and big pain for business.

But then appeared new small planes with much better fuel efficiency (787 and new modification of A-3xx, sorry forgot exact number), plus changed regulations accepted two-engined planes fly farther from reserve airport, so now any airport reachable with small plane and hubs concept slow dissolve.

Sure, still exist number of rare visited places, where just very few tourists appear, but for now their place in "post-hub world" is "under construction", nobody could predict how things will change in nearest months.

ICAO four-letter airport codes are hierarchical. The first one or two letters indicate the country, e.g.:

EDDF - Frankfurt (E: Northern Europe, D: Germany)

EKCH - Copenhagen (E: Northern Europe, K: Denmark)

VTBS - Bangkok (V: South/Southeast Asia, T: Thailand)

KJFK - New York JFK (K: USA)

Using ICAO codes, you can make sense of a written out route at a glance. Taking one of the examples from the link:

AUY TAH VLI BNE BKK CPH SFJ JAV JUV NAQ

becomes

NVVA NVVW NVVV YBBN VTBS EKCH BGSF BGJN BGUK BGQQ

I.e.: Vanuatu (a few jumps) -> Australia -> Thailand -> Denmark -> Greenland (a few jumps).

Make a connectedness matrix of airports x airports with 1 marking a connection and 0 marking no connection. You can now iterate over legs by multiplying the matrix with itself. Do this until all values in the matrix are zero. The previous iteration shows a 1 where the longest routes are.

Back when airline status was based on number of segments this would have been a useful tool if you could find the lowest cost route on a single airline group. For example "find the most number of segments at the lowest cost per segment using only One World flights"

My first thought was the Guam Island Hoper between Guam and Honolulu at 7 airports.

[added] 8 if you add MNL (Manila) on arrival to Guam

https://en.wikipedia.org/wiki/Island_Hopper

Is there a flight db preferably with price info?