why do you want to know the greatest endurance speed? Surely the more useful thing to know is the best fuel economy speed which, as someone else has posted, requires you to know Groundspeed and fuel flow rate. Staying airborne for hours on end does not guarantee you have covered the biggest distance on the tank full of fuel.
A theme which has been ellusive in this thread is that "endurance" is staying off earth as long as possible. It has very little to do with getting anywhere either promptly or efficiently. There are few reasons for staying off earth as long as possible, but the reasons which are valid, are important. Recreational fliers don't really think of an aircraft as a stationary (or nearly so) observation platform, but thats where endurance can be vital.
Endurance would be the same as "best fuel economy speed", which does not require that you know the groundspeed, as best fuel economy speed would be an airspeed, not a groundspeed. If you are flying for best fuel economy, you do not care about getting somewhere quickly, so groundspeed is not a factor.
Once you have determined the best
speed for endurance or economy, you
may be able to reduce the fuel flow by leaning. doing that will not affect the speed, but will improve the endurance/economy.
Agreed, staying airborne for hours does not guarantee that you have covered the greatest distance, because you wern't flying at enduarance to cover the greatest distance. You were flying at endurance speed to stay away from earth as long as possible, perhaps to observe for as long as possible (search), or wait for conditions on earth to change, so you can return (blocked runway).
You would want to know the best speed for endurance for an aircraft you fly, because that speed is one of the values which is important for safe operation in unusual circumstances. Sure, there are other more interesting speeds to know for your plane, but it does not diminish the importance of this one!
Pilot DAR