A good full synthetic can easily last you 10k oil change intervals and beyond. And yes, good implies there are lesser synthetics. I personally use AMSOIL SAE's oil line.
Honda's maintenance minder thingy has a range calculation that takes in factors of mileage, time, engine temperatures, and load; based on what their engineers have set as minimum and maximum for each specific engine. They stopped putting in maintenance mileage notes because the maintenance minder calculates the maintenance to a more personalized and reliable notification for that specific vehicle based on how it's driven.
With the above said, it all depends on several factors for how long you can go between changing your oil:
-is the vehicle constantly run to redline?
-is the vehicle constantly towing or have heavy load capacities?
-is there any modified work done or aftermarket components being used (e.g. intake, exhaust)?
-are there any known issues that would adversely affect the engine (e.g. blown head gasket, leaking seals, etc.) [This should obviously be fixed A.S.A.P. and I'm not endorsing otherwise. ]
If you can say no to all of the above then you're likely able to get 10k-25k out of a good synthetic oil. The unfortunate thing about that is, the maintenance minder's maximum is somewhere around 9k I believe, so it'll constantly prompt you to change the oil even though it may not be necessary.
The absolute best way to know for certain is to have the oil analyzed by a tester company for deposits and breakdown. Most individuals won't do this, since it requires a sample be sent off and costs about the same as changing the oil anyways, but it definitely is a good way to know exactly how the engine's health is.