By Jove, purpleinopp, your photo of the "crazy one" looks indeed a near-identical match to what I've got: tall, non-branching, flower-less! Tell me more: do you know it to be a variety of Cosmos, hence a legit wildflower? Have you had them in the past? And, the money question: do you know if they eventually flower? (I don't see a flower on yours.)
And thank you for the links. The Cosmos with which I am familiar in my garden appear very much like the photos from the first link (Cosmos sulphureus). And my new one's leaves look quite similar to the leaves in this photo I found in the second link (Cosmos bipinnatus).
I think I will let these plants carry on, and I will follow them with great interest to see if they do indeed flower.
Thanks again for your help, and for also showing me (indirectly) how much more helpful it is to take a photo of a plant isolated against a bright, neutral background.
This is a wonderful community.