If you go to a summer camp at a top college, and you put that on your other college applications, would it make you look like you want to go to that college where you had the summer camp?
No.
No. Be aware that apart from showing you did something productive with your summer, attending a paid summer camp anywhere is not going to necessarily impress admissions officers. There are some competitive summer programs that WILL impress AO’s, but generally, they are free. Having a job and volunteering are just as good as paid summer camps, and in many cases, better.
My D did a competitive engineering camp at a college. She had it on her common app and even wrote about it in one of the essays that was specific to her major, but, without naming the school. It didn’t seem to hurt her at all in the admission process.
Even if she named the school, it would not hurt.
AO’s are big boys and girls. The Yale AO will not get his/her nose out of joint if an applicant did Summer at Brown. Similarly, s/he will not assume that an applicant will not attend simply because they did a summer program elsewhere.
It seems people are really certain what AO will do regarding this issue. Any evidence?
I think it’s fair to say that an AO will be a lot more impressed with a kid who did TASP for free than with a kid who paid $6k for Summer at Brown.
You ask that like there is a bible to the admissions process where everything is set in stone and can be referenced as in a dissertation.
So no, no “evidence.” Just experience and what I’ve been told by AOs in the past. As is always the case on this site, one can accept or reject any response. Just like on the Hollywood Squares, “it is up to the players if a star is giving a correct answer or just making it up; that’s how they get the square.” 
Now the OP is basically asking, as an example if paying $7K to attend Summer at Brown would be viewed differently by a Yale AO compared to spending $7K on a similar program at Yale. If you want to discuss whether paying $7K to go to a summer program would be viewed differently than paying zero for a summer program or spending the summer working at Joe’s Pizzeria, then that’s a different question. But since that was not the question asked by the OP, let’s not go off-topic.
Not an AO, but have been doing interviews for Yale for over 25 years. A kid who tells me that he/she held 2 part time jobs over the summer, or volunteered at the local homeless shelter is getting a lot more points than the kid who did a paid for summer camp or a poverty tourism/mission trip. I don’t necessarily hold it against the kids who do the latter, but they better tell me some impressive stuff they got out of their privilege. This approach is consistent with the general evaluative guidelines we get from the AO with respect to what traits they want us to evaluate.
Colleges want to see you do something productive over the summer…
But working or babysitting your little siblings or taking a class or even paying to go to camp are all okay. But paying to go to camp is not as impressive as you might think…they are money makers for the college…but you get to learn some stuff and hang out with people who like things that you do.
I understand that going to an expensive camp isn’t super impressive BUT what if the student is considering a certain major and would like to use a paid-summer program to investigate how much they like it? How can that be a bad thing? We are considering a two week program at Wake or Davidson on Psychology next summer for our D21 with the purpose being to see how she likes the subject, not to impress colleges.
Getting off-topic, and not helping the OP, but I am of the opinion that doing anything during the summer is better than spending it on the couch playing Mortal Kombat 11.
@skieurope I didn’t think it was really off topic. Posters above are basically saying it’s a bad thing to go to a program you pay for…even if it’s at a different college. If the OP wants to go to an expensive summer program, there could be good reasons. I agree that the student should also do something else that summer.
@homerdog - I think you do what’s right for your child and don’t worry about college admissions.
“If you go to a summer camp at a top college, and you put that on your other college applications, would it make you look like you want to go to that college where you had the summer camp?”
No. In fact, some AOs will see it as a positive that you have experienced one college and now wish to experience another. It shows you are willing to step out of your familiar comfort zone. However, I have also read in college guides that some AOs consider expensive summer programs at top colleges to be somewhat elitist (similar to the kid who writes his college essay on his family’s three week long European vacation). Generally speaking, the free or low cost camps (which are also the competitive ones) or the camps that award actual college credit tend to be the most highly regarded.
This is an interesting thread. My S is going to an engineering program this summer that is fairly expensive but not prestigious. We’re letting him do this to help him figure out if he wants to be an engineer – NOT to help him get into a program. He’s also a summer camp counselor so he’s got more going on. I plan to encourage him to include it on his apps b/c it is how he spent his time — but don’t expect it to be an advantage.
I think outside a few really well known summer programs, these things are very very small blips on an AO’s radar…
I think using summer programs to explore possible majors is fine - especially in the early years of high school when the kid may be too young for most jobs and many volunteer activities. My kid took a class at Columbia - we chose that location because he could get there on public transportation, not because he had a future interest in Columbia. It was one of many, many computer programming related things he did over the years. I’m sure it didn’t hurt him to get into one of the top CS programs in the country, but since he did so many other things, I doubt it made any real difference.
It would not hurt your chance at college A if you had used another colleges’ summer camps. Importantly, demonstrate that you had used your EC time fruitfully. In fact, some schools even prompt you to talk about it, for example, one of the topics for short answers in Princeton’s application is “Tell us how you have spent the last two summers”. Perhaps through these summer activities, be it washing dishes at minimum wage, or attending a costly enrichment camp, you had developed your fine characters even further: perseverance; empathy; willingness to listen; bridge-building.
My youngest D attended a few pre-college programs in STEM as a way of investigating various topics. Over several summers she built a progression of hands-on experiences: a week-long bio-chem “camp” (got her hooked on being in the lab), a two week STEM residential program at Brown (It was her first time away from home, and she met a diverse group of kids from everywhere - loved the academic vibe but wasn’t impressed with dorms or food service!), and a three week biotech program with lots of lab work, including an overnight to Woods Hole Oceanographic Institute and numerous visits to biotech companies and hospitals in Boston/Cambridge. For the past two summers, she has been working in a college research lab (this year it actually pays a stipend - yay!). All in all, these experiences gave her confidence to pursue STEM and demonstrated her passion/focus to AO’s on her college applications.