What is the Turing degree of Kleene's O?
There is a talk given by Stephen Simpson, located here http://www.personal.psu.edu/t20/talks/pisa0104/talk.pdf, where he discusses the relative consistency strengths of various subsystems of second-order arithmetic $Z_2$. In particular, the theory $\Pi^1_1-CA_0$ denotes a subsystem of $Z_2$ whose set of true $\Pi^1_0$-consequences is of Turing degree $\emptyset^{(\omega^{CK}_{1})}$, which is the Turing degree corresponding to Kleene's $\mathcal O$. However, no proof is given, but I'm sure that one can be found in his bibliography for the talk (or perhaps even in his text on reverse mathematics http://www.personal.psu.edu/t20/sosoa/). 

The relevant information in the PDF is on page 29.