I think you mentioned previously there may be maintenance issues / extra checks required?
On a slightly different angle to that, is it possible to give us an idea of what kind of problems cause a "must be fixed before departure" vs "can carry to next airport"? I gather there would be some (minor) differences between aircraft and airlines?
eg. How many toilets can be u/s? What does the APU status need to be? Any flap restriction?
I guess as a pax i've got a list in my head of "stuff that's delayed me" - typically they fall (broadly) into engine, air conditioning and flap/throttle/mechanical issues whereas I've (not unreasonably) got basically no idea of what stuff is "ok" to "go without"?
There's a set of quite huge 'books' put out by all of the manufacturers, called 'Minimum equipment lists' (MELs for short). Individual airlines can make the list more restrictive, but not less so.
Toilets aren't airworthiness items, and I expect they could all be u/s and we could still take the aircraft flying. Obviously though, we wouldn't actually take any passengers with us in that case.
Flap/throttle/mechanical (which is a pretty broad brush) will generally stop the flight until fixed.
Engines must obviously be working (although some airlines indulge in 3 engined ferry flights), but, it may be acceptable for the autothrottle to be u/s, or a FADEC mode to be unavailable. A bleed valve could be locked closed, or perhaps reverse thrust locked out. The MEL will also come with time limits by which it must be fixed, and also procedures for both engineers and pilots. The MEL may also include performance penalties that need to be applied.
The requirement for the APU varies. The 380 and 747 don't need it at all, though it can be very inconvenient not to have one. The ETOPS twins will be restricted without the APU, as it's assumed that the APU would be available to help with the air and electrical loads if an engine were to fail.
With items like Air Data Computers, it may be possible to fly with one u/s, but the failure may only be allowed on one of the three carried, and so they might need to be physically moved around to ensure the failure is from the correct position.
The procedural changes and limitations thrown up by the MELs make the final decision to accept or reject their application subject to the Captain's approval. For instance, an MEL might say that the aircraft is not to be flown in icing conditions, and perhaps that seems like no problem to an engineer who is working in 40º in Sydney when despatching an aircraft to an equally hot destination...but, icing conditions will exist at altitude no matter how hot it is on the ground, so that MEL is basically always unacceptable.