When I teach irrational numbers, of course I have to include pi! (by the way, I have memorized 30 digits and also have pi earrings)
One way for students to grasp the idea of non-repeating, never ending numbers is to actually look at a "piece" of pi. This activity that I developed allows each student to get their own 100 digits of pi.
They color their piece and then I tape them all together for a visual representation-- we all look at it and are amazed that all of our work is only a small piece of the actual number (*mind blown*).
There are enough "pieces" for 102 students (the most I had one year when I did this activity). Enjoy!