I am tutoring a younger student in Algebra (primary school) and during the discussion about laws of addition and multiplication (commutativity, associativity, distributivity) I mentioned to her that a lot of the things we are learning about (polynomials, solving equations,etc.) apply equally if we use completely different operations to combine numbers as long as they obey the same set of rules. She seemed to buy this, but did not look satisfied without an example - and I wasn't about give a 13 year old a crash course in abstract algebra to use a more complicated example.
Later when I got home I realized that I didn't know of the existence of any group laws over the set of real numbers other than the canonical addition and multiplication laws. Do any exist? I would love to present something like this as a problem at our next lesson.