What is an example of: A is a subset of B and B is a subset C, but C is not a subset of A?

Hello!
Actually, such a situation is typical. If A is a subset of B and B is a subset of C, then A is a subset of C (any element of A is an element of B and thus is an element of C).
Therefore for C to be a subset of A, A and C must coincide. And for C not to be a subset of A it is sufficient that B has at least one extra element compared to A, or C  has at least one extra element compared to B. This is easy to achieve.
For example, let A = {1}, B = {1,2} and C={1,2,3}. Then all the conditions are satisfied: A sub B sub C,  but not C sub A.
Or A = NN, B = ZZ, C = RR.