Cantor's discovery of the existence of more than one infinity was a revolutionary change in human knowledge. He defined the notion of counting by bijections and showed that one can use infinities as numbers for counting mathematical objects.
I am planning a "Set Theory for Children" project to familiarize elementary school students with the basic notions of set theory.
There is no problem with giving an intuitive definition of infinity. Children in elementary school are already familiar with counting by natural numbers. One can convince them that the set of all natural numbers is infinite because it "never ends" and one can find a number larger than any given number by adding 1 to it.
But there are few concrete examples for infinities larger than $\aleph_0$. Also introducing basic transfinite arithmetic in a naive intuitive way can easily lead to paradoxes like Hilbert's Hotel.
Question. How can I explain the existence of infinities of different sizes and transfinite arithmetic so elementary school students will understand it?
Remark. I believe transfinite numbers and their arithmetic are as natural as finite numbers and finite arithmetic. The fact that they seem strange even for professional mathematicians has a root in our elementary school education and our incorrect/incomplete basic intuition about the notions of "number", "enumerating" and "arithmetic". So there is a natural way to teach these natural notions to students even in elementary school. We just need to discover such a teaching method.