The Oval Office is home to some of the most confidential and important conversations, but are they kept secret?
No, conversations in the Oval Office are generally not recorded. However, note takers may be present in important meetings, especially those with foreign leaders.
Presidents could choose to bug themselves if they like, but as Richard Nixon demonstrated for the ages, it's not such a good idea. No one is infallible. A complete record of every utterance of even the most virtuous president is bound to include something unflattering.
It also might have a chilling effect on the level of candour with which aides offer advice. The knowledge that their words are being recorded might cause a president's advisors pause about offering advice the president might not want history to hear. The ability of a president's staff to speak frankly is critical. When presidents are wrong, and they sometimes are, aides need to be able to tell them that without fear of tarnishing their legacy.
You can still listen to presidential recordings of the past. UVA's Miller Center has a terrific collection of fascinating audio recordings here.