I want to show that for all $n$ there is some collection of $n + 2$ circles such that two of the circles ($A$ and $B$) are tangential to each of the remaining $n$ circles (but not to each other) and each of the $n$ circles are tangential to both $A$ and $B$ but not to any of the other $n$ circles. My initial conception was to let $A$ and $B$ be two large circles (as large as necessary) that are extremely close to each other but not tangential. Then we draw one circle beneath $A$ and $B$ that is tangent to both of them. Slightly above it and in between the "AB crevice" we draw another circle. Then slightly above that we draw another. And so on and so forth until all the circles have been drawn. It seems to me in an intuitive sense that we could draw an infinite number of such circles, but I can$t seem to find a rigorous mathematical expression of this idea.
Note that if this idea is simply wrong, or if you have a simpler way to approach this problem, those would be acceptable answers as well.