The goals of the "American Dream" would usually be interpreted as suggesting that every individual has the right to achieve financial security and material success. The ideal of every American, according to the American Dream, is to improve his/her status in society by virtue of hard work leading to improved social standing.
Huck would want no part of most aspects of this definition of the American Dream. He had no wish to be involved with financial affairs, as he demonstrated in his dealings with Judge Thatcher.
I don't want it at all-nor the six thousand, nuther. I want you to take it; I want to give it to you-the six thousand and all...Don't you ask me no questions about it, please. You'll take it-won't you?...Please take it...and don't ask me nothing-then I won't have to tell no lies.
All the money meant to Huck was trouble from Pap.
Huck's vision of what he wanted from life did not involve hard work or accumulating wealth and material possessions. His preference was to be free to go where he wanted, when he wanted, without obligation to anyone else. That kind of detachment from society would be unthinkable to those striving to achieve the American Dream, but it completely satisfied Huck.
I reckon I got to light out for the Territory...because Aunt Sally she's going to adopt me and sivilize me and I can't stand it. I been there before.