We will discuss the Julia set of the map obtained in the limit of fixed points of the Feigenbaum equation as the order of the critical point goes to infinity. The main new result is that the measure of this set is zero. The proof proceeds by the standard route of martingale considerations, but a new element is that the relevant process for the limit map has infinite variance. This marks a difference between the limit Feigenbaum map and the maps of any finite order and makes estimating the measure of the Julia set much easier; on the other hand there are interesting probabilistic ramifications.