It could be turned to one. An elliptical sentence means that there's something in it that's missing, but within the context, can be understood. (Also note, it's "rigamarole" not "rigmarole") To use your example and turn it to an elliptical sentence, I would add a comma: "To begin with, the old rigmarole of childhood." Now, the sentence is elliptical because it can be used as a continuation of a thought that isn't stated completely. Here's a hypothetical context: "There are many things I didn't like about being a kid. [1] To begin with, (one of the things that I didn't like about it was) [2] the old rigamarole of childhood." The ellipsis in the above example is the phrase "one of the things that I didn't like about it was", which I put in parenthesis. [2] For the sentence to be a complete thought, you need this phrase. But you can get away with omitting it in the above example because it was referenced in the preceding sentence. [1]

Is this a quote? I see nothing wrong with the sentence although it does appear to have a literary flavour about it.

"To begin with the old rigmarole of childhood." It is elliptical because it is not even a full sentence. It is only a clause (noun or adverbial clause depending on how you read it). A sentence must have subject and predicate(descriptive part), and this phrase obviously does not. The author probably intended something like "I will relate my story beginning with the old rigmarole of childhood." By skipping the main clause "I will relate..." and turning "beginning..." into "To begin...", it was made more pithy and compelling (at least to me). And both "rigmarole" and "rigamarole" are accepted as correct ("dictionary.com" for example).