No commenter has noted that both chambers of the Alaska State Legislature begin every daily floor session with a prayer.  It's not necessarily a Christian prayer, though, and I don't think the presiding officer has anything to do with selecting that day's "chaplain."  I believer the Chief Clerk arranges for "chaplains."  If the chaplain is a no-show, the presiding officer asks a member to give the invocation, as Ramona Barnes asked Rep. Joe Sitton in '93 or '94.  Joe, who had, as was his habit, held down a barstool at the Alaskan the night before, was initially shocked to be asked, but then he delivered what might have been the best invocation ever offered in the AK House: "Lord, forgive us for what we are about to do to the people of Alaska today."