I am British, and I believe in American exceptionalism by one conception of the term, though I also believe in British exceptionalism, French exceptionalism, Russian exceptionalism, and a few others.
The US has done exceptionally well for itself -- at the start of the last century, it was a nobody; by the end, it was the undisputed superpower. This is indeed impressive. The US has the world's most powerful military, and by many measures the world's largest economy. Americans as a people are exceptionally friendly, outgoing and welcoming.
There are, however, plenty of ways in which the US is doing worse than other countries -- some immutable, some not. For example, the US simply lacks the institutional history to deal with questions of sovereignty crises as well as older nations like Britain or France. It's more susceptible to both extremist traditionalism and extremist revisionism, because of the lack of this institutional history.
The US also has huge crime problems, widespread poverty, a truly shocking healthcare system, poor education and an intractably corrupt local government system in some areas. These aren't unique problems, but that doesn't mean the US doesn't suffer from them -- and it's also true that some countries deal with them better.
Yet for all this I do tend to think of the US and Americans as somewhat headstrong and certainly impulsive but still good-natured children of ours. There's no doubt that of all our colonies, the US is the most powerful and influential.
I also fundamentally think that Americans' hearts are in the right place -- too many people think the US is just full of racist bigots and morons seeking to glass the entire Middle East and establish worldwide Christian American hegemony. I give you more credit than that. There are some of those sorts in your country, but then there's some of those sorts in every country. One shouldn't judge the country on that -- equally applied, I might add, to a place like Afghanistan, as troubled as it is.
Fundamentally I think the US is, in terms of national consciousness, trying to do right by the world, but a bit lost as to how to go about doing so. Like a child full of potential that rebels against his parents and doesn't listen to all of the lessons taught at school, I think the US at its roots is no longer rebellious (clearly not) and has taken a seat at the table of Western nations that have the power to shape the world, alongside countries like Britain, France, Germany, Canada, etc. The US may have missed out on a few lessons Canada got from its parent nation, but the US learned some valuable lessons of its own.
SO! Yes, I do believe in American exceptionalism, if by exceptionalism you mean "the belief that the US and the American people, while not perfect, have a distinct and good role to play in the world, and their struggle is our struggle, and vice versa."