surveys United States colleges and universities each year about their admission policies. Question C3 asks if a high school diploma is required for undergraduate admission.
Harvard
http://www.provost.harvard.edu/institutional_research/Provos...
does not require a high school diploma for admission.
Neither does Princeton.
http://registrar.princeton.edu/university_enrollment_sta/com...
Nor does Yale require a high school diploma.
http://www.yale.edu/oir/cds.pdf
MIT has long reported that it does not require a high school diploma for admission.
http://web.mit.edu/ir/cds/2010/c.html
http://www.mitadmissions.org/topics/qanda/questions_and_answ...
There are other colleges that explicitly say in their Common Data Set filings that they do not require a high school diploma for admission. Moreover, homeschooling is widespread in around the world,
http://learninfreedom.org/homeschool_growth.html
and all of the most famous and most desired colleges and universities have admitted homeschoolers,
http://learninfreedom.org/colleges_4_hmsc.html
who often have "home brew" transcripts (as my oldest son did when he applied for his undergraduate university studies last year).
Lacking a high school diploma issued by a government-operated school is not a barrier to admission to any of the better colleges or universities in the United States, if the applicant is well prepared for higher education study.
After edit: I'm amazed that this thread has not yet mentioned pg's essay "What You'll Wish You'd Known,"
his advice to high school students about how to use their time meaningfully. High school students who take this advice to heart can get into a good college with good financial support if they want to, or pursue some other challenging personal goal if they would rather do that.