@neeple , here are a few threads that asked the question of which Elite universities provide Merit Aid. There are many more similar threads like this, but these should give you a good idea of what to expect when seeking Merit Aid from elite universities.
Merit Scholarships for ACT 36, top 5% of class: http://talk.qa.collegeconfidential.com/financial-aid-scholarships/2197334-merit-scholarships-for-act-36-top-5-of-class.html (10 responses to date)
Looking for advice in Merit aid for a top 1% student: http://talk.qa.collegeconfidential.com/financial-aid-scholarships/2154331-looking-for-advice-in-merit-aid-for-a-top-1-student-p1.html (1600+ responses to date)
Merit Scholarships for 33 ACT and top 10 percent class rank (2016 thread, you will need to verify these recommended offers): http://talk.qa.collegeconfidential.com/financial-aid-scholarships/1910341-merit-scholarships-for-33-act-and-top-10-percent-class-rank.html (21 responses)
Merit Aid College Search Help: http://talk.qa.collegeconfidential.com/financial-aid-scholarships/2192557-merit-aid-college-search-help.html (15 replies)
Merit Scholarship From Top Colleges: http://talk.qa.collegeconfidential.com/financial-aid-scholarships/2194728-merit-scholarship-from-top-colleges-p1.html (25 responses)
The last one asks the following question, which I think is what you want to know specifically: “What are the real odds of getting a substantial merit scholarship from a top college like Ivies or other elites in similar tier? Is it possible to get at least 50% or more of tuition free from such colleges if you don’t qualify for aid or athletics?”
PS: Some of the threads I referenced include information for NMSF students.