I believe the exception is impeachment. Articles of Impeachment can be brought against any civil official. There's nothing stopping Congress from impeaching and removing the Ambassador to China, a Supreme Court Justice, or pretty much anyone else operating under the authority of the federal government.
It's rare for this to happen of course, but this list of impeachments is interesting reading: https://history.house.gov/Institution/Impeachment/Impeachmen...
As recently as 2010, a district court judge was impeached, convicted, removed from office and barred for perjury.
Also, as far as I'm aware, it is within the power of the Presidency to pardon themselves. If by some chance someone were elected President who had previously been convicted of a federal crime, they can pardon themselves of that crime. In theory, they could even be in prison at the time of election! I would love to say that's unlikely in the extreme, but in 2015 I watched Joe Morrissey win re-election to the Virginia House of Delegates while serving a prison sentence: https://www.reuters.com/article/us-usa-virginia-election-idU...
There are a couple of opinions that say charging a sitting president with a federal crime would be unconstitutional, but I don't believe they're technically legally binding, nor do I understand that it was ever been decided in court.
Impeachment isn't a criminal process. It's a political process that rests entirely within the Legislative branch. So it's not as much an exception as a wholly separate proceeding.
There are plenty of examples of embarrassing pardons, but it's not a rule of law failing.