In the second section, we consider several history-dependent sequences which have attracted recent interest. We primarily study the Ulam-Kac adder, a sequence for which very little is known explicitly except for its first two moments, which have been computed in some generality. Our main contribution is to compute the asymptotic behavior of all moments and obtain bounds on their size. We also provide a semi-analytic formulation of the moment problem which allows them to be computed directly. We discuss many combinatorial interpretations of this sequence and others which, in particular, lead to a novel formula for the first passage times of a related sequence. Connections of these sequences to other areas of mathematics and physics are explored.