Here's proof that top-ranked universities really do welcome homeschooled students. I explored the web to see what’s expected of homeschool students applying as first-year undergraduates, and what I found was both encouraging and informative. For high-achieving students, begin planning early for high school in order to meet admissions requirements for these elite schools.
University of Chicago
Massachusetts Institute of Technology
University of Pennsylvania
California Institute of Technology
Johns Hopkins University
Washington University in St. Louis
University of Notre Dame
Attending this type of university might not cost as much as you may think! Many of these schools are "need blind", meaning that acceptance to the school does not depend on the family's ability to pay. Basically, if a student is accepted to the university, the family would pay an amount corresponding to demonstrated financial needs; the rest is covered by grants, work-study, and scholarships, without taking out loans. For participating universities, a Net Price Calculator can be found on the College Board website. Check the Financial Aid sections of individual university websites for more information.
Many universities, both public and private, use The Common Application with the Home School Supplement to apply online for admissions. It's a good idea to print out these forms now and review them for future reference.