I am following the list with great interest. As a community college math teacher, I teach many courses "below" calculus, but I try to inject the concept of a limiting process wherever I can - so that when they arrive in calculus they are prepared for a formal derivation.
As a few examples:
1. The formula for the area of a circle can be justified by disecting the circle into wedges which can be set down in alternating orientations to approximate the shape of a square. The smaller the wedges the, the better the approximation - and if the wedges are infinitely thin, then the approximation becomes exact.
2. The Babylonian algorithm which converges to the root of a number. A nice example of digit doubling as well.
3. Other feedback loops with iterative function.
4. The Koch curve (snowflake). This one is nice since it shows that, when the limiting process is complete, you have a new and marvelous object with qualities unlike any of the approximations.
5. In developing the formula for natural growth we have the limit of (1+1/n)^n as n->inf, numerical data are used to justify the definition of e.
If a student can digest and come to appreciate these, then they are ripe for appreciating what it really means for a curve to be smooth.