We define two successive extensions of Kleene's O using infinite time Turing machines. The first extension, O^+, is proved to code a tree of height λ, the supremum of the writable ordinals, while the second extension, O^++, is proved to code a tree of height ζ, the supremum of the eventually writable ordinals. Furthermore, we show that O^+ is computably isomorphic to h, the lightface halting problem of infinite time Turing machine computability, and that O^++ is computably isomorphic to s, the set of programs that eventually write a real. The last of these results implies by work of Welch that  O^++ is computably isomorphic to the Σ_2 theory of L_ζ, and by work of Burgess that O^++ is complete with respect to the class of the arithmetically quasi-inductive sets.