If the class has a compare method that doesn't work for my current problem, do I subclass it and create my own maxstr?

If you are referring to my original post, then yes. The Max class abstracts the algorithm of calculating a maximum of objects - the subclass implements the comparison in the appropriate context. Im not suggesting this is the best OO solution, just one that is like the Higher-Order equivalent.

If you meant "Foo.compareTo() doesnt compare Foos like I want", then the solution depends on (among other things) how much internal access you need to compute the comparison. There are quite a few ways this could be approached but to bring it back to my original point, they are mostly structured and organized - not hidden un-garbage-collected variables and blind function dereferencing.

What if each object needs a unique compare function?

Then what are you comparing it to? I would have thought similarity was a prerequisite for comparison.

What if these functions aren't known at compile time?

Closures are compiled too. That is, whether you use gt or > (or whatever) is decided at compile time - there is no runtime dynamic there.

What if there are so many functions that it's difficult to give them all unique and useful names?

They are all called 'compare' - different classes do different things when it is invoked.

update: clarification - Im not sure Im addressing your questions?