Updated, Santa Clara and Carnegie Mellon added:
Required for all frosh applicants:
Caltech: http://www.admissions.caltech.edu/content/academic-preparation-requirements
Harvey Mudd: https://www.hmc.edu/admission/apply/first-year-students/eligibilty/
Olin College of Engineering: http://www.olin.edu/admission/apply/
Required for some frosh applicants:
Cornell, Dyson AEM and Engineering: https://admissions.cornell.edu/pdf/freshman-requirements-chart
Duke, Pratt School of Engineering: http://admissions.duke.edu/application/overview
Recommended for all frosh applicants:
Cal Poly SLO, +125 MCA points for each semester (up to four) of advanced math (first two semesters of advanced math are precalculus, third and fourth are calculus): http://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1014&context=statsp and https://admissions.calpoly.edu/applicants/freshman/criteria.html
Kenyon: “ideally up to calculus” “recommended” http://www.kenyon.edu/admissions-aid/how-to-apply/preparing-to-apply/
MIT “recommendation”: http://mitadmissions.org/apply/prepare/highschool
Minnesota - Twin Cities “preferred”: https://admissions.tc.umn.edu/admissioninfo/fresh_overview.html
Stanford “most competitive freshman applicants often have … four years of math (including calculus)”: https://admission.stanford.edu/apply/selection/evaluate.html
Wesleyan “recommended”: http://www.wesleyan.edu/admission/apply/deadlines.html
Recommended for some frosh applicants:
Carnegie Mellon, College of Engineering, Mellon College of Science, School of Computer Science, Tepper School of Business “mathematics should include … preferably calculus”: https://admission.enrollment.cmu.edu/pages/academic-requirements
Columbia, School of Engineering and Applied Science “preferably through calculus”: http://undergrad.admissions.columbia.edu/ask/faq/question/2407
Northwestern, McCormick School of Engineering and Applied Science “calculus preferred”: http://admissions.northwestern.edu/faqs/high-school-courses/index.html
Penn, Engineering and Wharton “would like to see”: http://www.admissions.upenn.edu/apply/whatpennlooksfor/hsprep
Princeton, for engineering majors “if possible, we expect students will complete”: https://admission.princeton.edu/how-apply/academic-preparation
Rensselaer Polytechnic Institute “recommended”: http://admissions.rpi.edu/undergraduate/admission/freshman/requirements.html
Santa Clara “for natural science, business, and engineering applicants, readiness at the level of calculus is expected”: https://www.scu.edu/admission/undergraduate/first-year-students/
Stevens Institute of Technology, for engineering, computer science, applied science, and quantitative finance majors “recommended”: https://www.stevens.edu/admissions/undergraduate-admissions/how-apply/first-year-students
USC Viterbi School of Engineering “recommended” “important preparation”: http://viterbiadmission.usc.edu/apply/#firstyear
WUStL, Architecture, Business, and Engineering “strongly recommended”: https://admissions.wustl.edu/apply_site/Pages/First-Year-Student-Requirements.aspx
Forum member heard from an admissions representative that calculus is required, but that is not stated on school’s web site:
Georgia Tech: http://admission.gatech.edu/freshman/academic-preparation
Michigan, Engineering: https://www.engin.umich.edu/admissions/undergrad/first-years/admissions/