Sup

What is the "sup" function?

As stated in the wikipedia article:
"sup(S) is defined to be the smallest real number that is greater than or equal to every number in S."

What does it do?

Read above.

How is it used?

Sup is not a specific algorithm - it is just a definition/function.

What is the point of using it?

If you need the smallest real number that is greater than or equal to every number in set or subset, you use sup.

How does the Legendre transformation work?

http://en.wikipedia.org/wiki/Legendre_transformation

Can you please provide examples (preferably with functions that variables can be plugged into)?

Google shows the second link to be this:
http://www.student.fizika.org/~nnctc/legendre.pdf
which answers things nicely.

What does p represent?

A function or a number - anything that really makes sense within the context. If you wonder what it usually means, look up the answer above.

I know that in transformations, f(x) is usually mapped onto a different variable, but in this particular transformation, doesn't really seem to be doing anything to f(x), because it's external to f(x).

Above. Besides, f(x) might be a function, so not really. But in usual application, as mentioned before, look above.

Also, if the sup function is the supremum function, does that imply that the curve is full of maxima?

No. A function doesn't imply anything.

Out of curiosity: If I differentiate x(t) = cos(ω*t), ω>1, an infinite number of times, does it eventually diverge?

It doesn't diverge, because the infinite derivative is not defined for that function.

In other words, if x'(t) is the velocity, x''(t) is the acceleration, x'''(t) is the jerk, ... and the infinite derivative is the infinite change in some quantity of motion, does that imply that a simple oscillator can have an infinitely large change in some quantity describing the motion?

This is a whole other question, because you're now taking a mathematical aspect and assuming it makes sense to apply it in a physical situation as a "description" of the system. As for your question, the infinite derivative of the position x(t) does not make sense and is not used in modern physics - and no, as the infinite derivative is not defined, we cannot do so.