The code is the more desirable/selective the school is, the fewer the scholarships, if any, and the greater the competition. So if you want to maximize your chances, you have to go further down the rung in selectivity/ ranking. So many statistically equal candidates will be vying for these scholarships in terms of test scores and gpas that I think even more rides on the essays (including specialized scholarship essays), EC’s, awards and honors than even for the most selective schools’ admissions decisions. Further, while these are merit scholarships, I don’t doubt other holistic measures come into play like family circumstances. For state schools, residency may be a stated or unstated factor.
You can look at some of the named scholarships, such as
JHU: https://apply.jhu.edu/hodson-scholars/
Duke/UNC: https://robertsonscholars.org/
Stamps (various colleges): https://www.stampsscholars.org/
As you can see, the exact “sort” of student varies a bit, but basically you should be a top student who is clearly on their way to changing the world 
This shouldn’t be very difficult for you to look up. The top 25 will be immediately reduced to 17 when you remove the Ivies. Go to the financial aid website of each school to learn what scholarships they offer. The eligibility requirements should be listed there. You could go down the rankings list a dozen schools at a time until you find schools that are in the price range you want to pay.
What’s your child looking for in a school?
There are a number of other schools in the top 25 that give zero $$$ in merit aid. Stanford, MIT come to mind.
@1Rubin what colleges is your kid considering? And what are the kid’s stats? There are some higher ranked schools that give merit aid…but these merit awards are most highly competitive. Vandy has some merit awards. UVA has the Jefferson Scholars. There are some others, but as noted…the competition for merit awards at these schools is more so than admissions which are low low %age of applicants as well.
But the good news is…if your student really is competitive for admissions at these elite schools…there are plenty of places where a strong applicant like that could get significant merit aid.
Stats? Possible major? Geographic location? Your price point annually out of pocket? Give us this info and I’m quote sure you will get some good suggestions.
If you really want this information, consider taking the time to review the most recent Common Data Set information for each of the colleges that may be of interest. The financial aid section will be a little more than 1/2 way through. Once you find it:
(1) Compare the line for Institutional Scholarships/Grants (a) Need Based; against, (b) Non-Need based. The total dollar value of non-need based aid excludes non-need based aid used to meet need, so it is a “merit-only” number for all undergrads (freshmen through seniors);
(2) Go to the next page, which should have a section labeled “number of enrolled students awarded aid” and then: (a) look at the total number of first time full time freshmen; (b) skip over to the next chart showing the number of those freshmen “who had no financial need an who were awarded institutional non-need-based scholarship or grant aid.” That gives you the raw number of freshmen receiving any merit aid and allows you to determine what % of freshmen receive merit-only aid.
(3) The next line should show the “average dollar amount of institutional non-need-based scholarship . . .” Since it is only an average, this number does not necessarily reflect a typical merit scholarship.
(4) Compare the information you looked at in #2 and #3 with the column providing the same information for all Full-time undergrads. If the average dollar amount is about the same and the number of freshmen is 25% of the number of all undergrads with merit scholarships, that suggests the college’s merit scholarship practices are stable. If there was a big drop in the number receiving merit or the average dollar amount for freshmen, that might suggest they are moving closer to a need-based-only model.
I want to thank all posters for sharing their knowledge and tips.
How do you determine this information?
That is insane (unless the family has significant millions).