Here's another idea. Question to the class: Starting with a square, how many times do you think you can cut the figure in half and then remove a half and then repeat the process on the figure remaining? The students may suggest a finite number (say 15). The teacher gives each student a square sheet of paper and has them actually go through the process. Eventually, they end up with a piece so small that it seems impossible to cut in half. Now, the teacher comes back and says "imagine if you were really really tiny and had a tiny pair of scissors, do you think you would be able to cut the figure in half again." The students may suggest yes, and suggest the idea that as long as they could keep shrinking themselves and their scissors, there will always be a piece to cut in half. The teacher can have the students represent their work with numbers: How much area of the original square remains after each cut? This example introduces students to the idea of infinity, infinite series, and one-to-one correspondence while employing their current knowledge of areas (each piece is some fractional area of the original) and geometric figures (the figures oscillate between square and rectangle).
I remember in third or forth grade they had use cut out a triangle, then cut all the corners off the triangle and put their tips together. No matter what triangle you made when you put all the corners together they made a straight line. I distinctly remember that glimmer of insight I saw. I still didn't understand why, and almost didn't believe it. But it was right there in from of me.
I don't think I saw proofs again till high school pre-cal. I had a very good math/physics/comp sci teacher all through high school and I remember he showed proofs for nearly every formula we used, but only demanded we learn a couple. I think in that in Manitoba, Canada it is expected that all grade 12 pre-cal students be able to prove the law of cosines; it often comes up on provincial exams. Also, I noticed someone here said they didn't do proof till after intro calc? Well I am not sure about anywhere else but at my university on the first day of intro calc they said you will be expected to be able to prove every theorem used in the course, and believe me they didn't shy away from that promise in the final exam! Every university math test I have ever taken is at least 50% proofs.